Math Skills for Introductory Economics by qym17251

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									      Math Skills for
Introductory Economics

     Geometry


            Rick Fenner
           Jerry Evensky
     Department of Economics

       Charles M. Spuches
      lnstnrctional Designer
                       Table of Contents

Introduction         ..................................................................................    3

How to use this book               ...................................................................     4

Variables      ........................................................................................    5

Graphs     ............................................................................................    6

Slopes   .............................................................................................. IS
Nonlinear Relationships                    ........................................................... 26
Summary         .................................................................................
                                                                                                33

Review Examination                  ..................................................................34
Review Examination Answers                           ................................................     37




                                                                                                           Geometry I 1
                                                                                                                          n13
Introduction

This book has been designed to provide you with the geometry skills that you
will need in the Introductory Economics courses. Mathematics plays an
important part in economics and serves, in fact, as a basic means of
communicating what is happening in an economy and why. Experience has
shown that students who lack command of certain basic math skills will usually
have significant problems passing a basic economics course. This material, when
mastered, will facilitate your study of Introductory Economics.
Specifically, at the completion of this unit you should be able to:
   1. Identify different points on a graph using coordinates.
  2. Match a graph of a straight line with the appropriate equation.
  3. Match an equation of a straight line with the appropriate graph.
  4. Calculate the slope of a straight line
          a. from the equation
         b. from two points
          c from the graph
  5. Determine whether the slope of a straight line is positive, negative, zero,
      or infinite.
  6. Identify the point of intersection between two lines on a graph.
  7. Determine points of tangency on a curve and use them to calculate the
      slope of the curve.
  8 Determine whether a curve (or portion of a curve) has a positive,
   .
      negative, or zero slope.
 9. Ident~fy     maximum and minimum points of a curve.




     IMPORTANT Before beginning, you should have a pencil and several
     sheets of work paper available.




                                                                  Geometry 1 3
       o
  How T Use This Book

 To help you use this booklet in geometry, each section begins with an
 explanation of the.particular geometric concept that will be covered. The concept
 will be followed by an Example question that will be worked out for you step-by-
 step. Make sure you understand the process before you move ahead. If necessary,
 look back at the text for important rules or definitions.
Following the example will be a section called Practice. Work out the questions
asked. Try to do this without looking back at the text. Answers to the Practice
problems will be found on the next page. If you get all of the practice questions
correct, continue on to the next section. If any of your answers don't match, go
back and find out why they are wrong. If necessary, go back to the text. Don't go
on with further sections of the book until you can correctly answer all of the
Practice questions.
In the back of the booklet you will find a summary of the skills ipg. 33) you
should possess when you have completed this unit. Following the summary is a
review test (pg. 34) which you should complete withoutgoing back to the text.
The answers follow the test.
All set? Let's begin.




4 1 Math Skills for Introductory Economics
 Variables

 In economics, characteristics or traits such as prices, outputs, income, etc., are
 measured by numerical values. Since these values can vary (e.g., price can
 change from $10 to $20), we call these characteristics or trails "variables." Variable
 is the generic term for any characteristic or trait that changes.
 We express a relationship between two variables, x and y, by stating the
 following: The value of the variable y depends upon the value of the variable x.
 For example, the total price of a pizza (y) depends upon the number of toppings
 (x) you order.
We can write the relationship between variables in an equation. For instance:

is an example of a variable relationship between x and y. If we know that x and y
represent variables, what are a and b? They are constant. In other words, they are
fixed values which specify how x relates to y. In our pizza example, "a" is the
price of a plain pizza with no toppings and "b" is the cost of each topping.
If we draw a picture of this particular relationship between x and y, we'll see that
all the combinations of x and y that fit the equation, when plotted, form a
straight line.


       y = final cost
       a = price of plain pizza
       b = cost of each topping
       x = number of toppings
If plain pizza (a) is $3.00and cost of each topping (3)is $.60, we pet :


     Final Cost           Plain        Cost of Each Toppinq    Number of TopPinQS
     $3.00              $3.00                 $.60                    0
       3.60               3.00                 .60                    1
       4.20               3.00                 .60                    2
       4.80               3.00                 .60                    3
        5.40           3.00                 .60                  4
We can take this information and put it into picture form. These pictures are
called graphs.



                                                                           Geometry 1 5
 GRAPHS

 Groping Points
 In economics, we often use graphs to give us a picture of the relationships
 between variables. Below is an example of a graph with a series of points drawn
 on it.




The axis are labeled x and y. In economics we will usually give the axis different
names (Price and Quantity, for instance), but for our mathematical rules, always
think of the horizontal axis as the x-axis and the vertical axis as the y-axis. In our
pizza example, the y-axis is the total selling price and the x-axis is the number of
toppings.
In the graph above,
       Point A is the origin or point (0,O).
      Point B is on the x-axis and is labeled (3,O);
      3 is the x-coordinate and 0 is the y-coordinate.
      Point C is on neither the x- nor y-axis, but is in the interior. Its x-
      coordinate is 1 and its y-coordinate is 4, so point C can be labeled (2,4).
      Point D is a s in the interior. Its x-coordinate is 4 and its y-coordinate is 2,
                   lo
      so point D is (4,2).
Take a minute to review this once again ...it's important.




6 1 Math Skills for Introductory Economics

                                                                                   '     n13
 Example.




       What point is on the y-axis?               A
       What point is labeled (20,60)?             D
       What point(s) have a y-coordinate o 30?
                                          f       B&C
       What point has the largest x-coordinate?   E
Example:




       Which point is (400, loo)?                 D
       What is the y-coordinate o point C?
                                 f                0
       Which point is on the x-axis?              C
Now let's see if you are still with us.


                                                        Geometry 1 7

                                                            '    018
 Complete the following practice questions on a piece of scrap paper and, when
 you are done, turn to the next page and check your answers.

  Practice 1 :
            1




 (a)    Which point is (0, 2)?
 (b)    What is the y-coordinate of point C?
 (c)    How many points are on the x-axis?
 (d)    What is the x-coordinate of point A?




8 I Math Skills for Introductory Economics
                                                                        .   ?
                                                                                 01.3
        o
Answer t Practice # I :
  (a) Which point is (0,2)?                         A
    (b) What is the y-cootdinate of point C?        1
    (c) How many points are on the x-axis?          (don't forget point D)3
    (dl What is the x-coordinate of point A?        0

Matching a Graph of a Straight Line with the
Appropr~ate Equation
If you recall, in our pizza example we have the following final costs based on the
equation (y = a + bx).

      Fino1Cost          Ploin        Cost of Each Toppina    Number of Toppinas
      $3.00        1    $3.00                $.a0                    0




The following is a graph of this equation (y = a + bx) where a = $3.00 and b = $.60.


                             Y
                         7   -
                         6   -



                   E     2-
                         1   -
                         0        1      7       9      v
                                                                X
                             0    1      2       3      4      5
                                    N U W R Of TOPPINGS




                                                                         Geometry 1 9
 Here is another example of the graph of an equation. Below is the graph of the
 equation (y = x + 30).




We can prove to ourselves that this is the graph of the equation (y = x + 30) by
checking to see that any two or more points on the line satisfy the equation.
Point A, for example, is (0,30); the x-coordinate is 0 and the y-coordinate is 30.
Plug (0,30) into the equation (y = x + 30).


These coordinates work in the equation! Now, let's try point C, which is (20,501.


                                       r
Again, it works! Point E is (40, 70). T y these coordinates in the equation.
       (70=40+30)or70=70
It works!
We have now tested three points on the line and they all satisfy the equation.
You must always try at least two points to be sure you have the correct equation
for the line.

Example:
Consider the following graph of an equation.
                       Y




10 1 Math Skills for Introductory Economics
 Is the equation of line AD :



 (Note: A line can be named by using the two end points, in this case, A and D.
 The line over AD signifies that we are referring to "line AD")
 Let's substitute some of the points on the line into each equation, as a check.
 First, try equation y = 17- 6x using point B, where point B is (2, 5); then try Point
 Dl which i (6,3).
             s
        Point B is (2,s)                  Point D is (6,3)




        5 = 5 This looks good.            3 = -19 This is obviously incorrect.


So, y = 17 - 6x cannot be the equation of the line. Now let's try our second
equation, y = 6 - x
       Point B is (2,s).                Point D is (63).




       5=5                               3=3
Points B and D check out, let's d o one more.
       Point C is (4,4).




Since all points check out, we know that the equation of the line is y = 6 - X x .



           I        REMEMBER: Always    check at least two points.
                                                                         I
                                                                         I
 Matching an Equation with the Appropriate Graph
 Example:




Which line, W or         , is the graph of the equation y = K x + 2O? To answer this
question, we do the same thing as before; we substitute the coordinates of
different points into the equation to see which line fits.
Let's try point B (20,401 which is on AD .




This is not correct. Therefore,           can't be the graph of the equation. Now let's
try some points from      n.
       Point E is (30,30)                     Point F is (60,40)
       y = H x + 20?                  -       y=Hx+20?
        30 = K(30) + 20                       40 = W(60)+ 20
        30=10+20                              40 =     20
        30 = 30                               40=20+20
                                              40=40
Both points from line         work in the equation, therefore,        is the correct
graph of y = Kx + 20.



12 1 Math Skills for Introductory Economics
Now, try these two practice examples. Again, you will find the correct answers on
the next page but, please, no cheating!!!

Practice #2:
What is the equation of the line shown in the graph below?




Practice #3:
Which line is the graph of the equation y = 5 + gx?




                                                                 Geometry 1 13

                                                                                 034
          o
 Answer T Practice Questions
   Practice #2: The correct equation is (b) y = 2x + 5.
     Practice 1 3 The equation Is represented by AD.
               1:
 If you got either answer wrong, review the materials before going on.

 A Final Note on Y - Intercepts
 Remember that the formula 'for any line is:


The y-intercept is the value of y when x is equal to zero. Note that if x = 0 then
                                                                             ,
y = a. When we use graphs, we call this point (0, a) the y-intercept. The y-intercept
tells us where the straight line of the graph intersects the y-axis. In the equation
above, the line intersects the y-axis at a. In our pizza example, the y-intercept occurs
when there are no additional toppings (x=O);the y-intercept is at a, which is the price
of a plain pizza.




14 1 Math Skills for Introducton/ Economics
 Slopes

 The slope is used to tell us how much one variable (y) changes in relation to the
 change in another variable (x). This can be written as follows:
                                         change in y
                                 slope =
                                         change in x

 Calculating the Slope from the Equation
 The equation of a line is given by:
                                variable
                                      \



Notice that b is the slope of the line. Let's label the equation for our p i u a
example:
                              total           cost
                              cost of          pcr
                              pizza

                                  \  y=a+hx  i'"
                                 bf plain      of
                                 pizza         toppings
Notice that the slope of the line tells us how much the cost of a pizza changes as
the number of toppings changes. Each additional topping raises the cost by $.60.
The equation for the total cost of a pizza is:


Where the y-intercept is $3.00 and the slope of the line is $.60. Now, let's look at
a few more examples.




                                                                        Geometry I 15
 Examples:
     y=20+30x                 y -intercept is 20
                              slope is 30

        y=4-lox-              y -intercept is 4
                              slope is (-10)


        Y=%x+%              y -intercept is )$
                            slope is %
When you are sure you understand how to determine the y-intercept and slope
of a line from its equation, do the practice exercises below. The answers to these
are given on the next page.

Practice #4:
What are the slopes and y-intercepts of the following equations?
      (a) y=$$x+6




 6
1 1 Moth Skills for Introductory Economics
                                                                                     02'7
          o
 Answers T Practice 54:
                                 Slope        Y-intercept




If all of your answers are correct, proceed to the next section. If you missed any,
review the text and example and try the practice items again.

Calculating the Slope from the Coordinates of Two Points
In the equation 4 = 200 + 30x, we know that the slope equals 30. This means that
for every one unit change in x, y will change by 30 units. But what if we are not
given the equation of a line? Can we still figure out the slope? If we at least know
two points on the straight line then, yes, we can determine the slope.

Example:
Let's say that points (8, 15) and (7, 10) are on a straight line. What is the slope of
this line?
                                           change in y
                                  slope =
                                           change in x
First, let's determine the change in y. To do this, we must choose one of the two
points to be our starting point. It doesn't matter which we choose, so let's take
(8, 15) as our initial point. The y value of this point is 15. (Remember that points
are always written in the form (x, y), that is, with the x value first and the y value
second).
The value of y changes to 10 in the next point (7,lO) so the change in y between
the two points is 5 (15 - 10).
                                                       5
                         So far we have slope =
                                                  change in x
Now let's measure the change in x. It's very important to use the same starting
point. The value of x at the first point is 8 and it changes to 7, so the change in x
is8-7=1.
We can plug the changes in y and x into the definition of slope:
                                  change in y 5
                          slope =             =-=5
                                  change in x I




                                                                        Geometry 1 17
We said that it really doesn't matter which point we choose as our starting point;
let's prove it. Let's take the same two points ((8, 15) and (7, lo), but this time we
will calculate the slope using (7, 10) as the initial point.
        The change in y = 10 -15 = -5
      The c h a n g e i n x = 7 - 8 = - 1
Put them together:
                                             change in y    -5
                                 slope =
                                             change in x
                                                           =-=s
                                                            -1
and you get the same answer as before!

Example:
What is the slope of the straight line connecting points (5, 3) and (3, B)? Let's use
(5,3) as the initial point.
       Change i n y = 3 - 8 = - 5
       Change in x = 5 -3 = 2
                                            change in y = -= -2-I
                                                          -5
                               Slope =
                                            changein x 2        2
You should now be ready to try a few practice exercies. Below are four exercises
where you can practice calculating the slope of a line from two points on the line.

Practice #5:
What is the slope of the line connecting each of the following sets of points:
   (a) (62) and (5,1)?


   (b) (03) (8,5)?
           and


   (c) (4,l) and (1,9)?


   (dl (6,6) and (3,9)?




                               Economics
18 I Moth Skills for lntraducto~
            o
 Answers T F:actice 85:
 The slopes of the lines connecting these points are as follows:
    (a) (6,2) and (5,l) -      =1
             and
     (b) (03) (83)                =4 =)(
    (c) (4,l) and (1,9)      ==-%
    (d) (6,6) and (3,9)         = =) = -1
 If you had a wrong answer, take a few seconds to find out why. ONWARD!

 Calculating the Slope from the Graph
 If we have an equation such as y = 2+Hx, we know that the slope is equal to        K.
But what if we only have the graph of the equation? How can we find the slope?
We can easily calculate the slope by picking out two points on the graph of the
equation and using our formula for slope.
                                        change in y
                                slope =
                                        change in x

Example:
What is the slope of this line?




Point B on this graph is (2,3) and point C is (4,4). Let's use C as our starting point.
      Changeiny=4-3=l
      Changeinx=4-2=2
      Slope =   K

                                                                       Geometry 1 19

                                                                                .         030
 Example:
 What is the slope of line   x?




Let's use point A (0'30) and point C (20,O). We'll start with point A.
        Change in y = 30 - 0 = 30
        Change in x = 0 - 20 = -20
        Slope=   %=-%
Now, practice calculating the slope of a line. Below are two practice exercises.

Practice #6:
What is the slope of line    x?




2 I Moth Skills for Introductory Economics
 0
                                                                                   031
Proctice #7:
What is the slope of line   m?




                                 Geometry 1 21
          to
 A,~swer Practice Exercises
    Practice #6: The slope of line AC is -1.
     Practice #7:      The slope of line     W   is 3.
 If you had either question wrong , review this section.

 Determining Whether the Slope of a Line is Positive,
 Negative, Infinite or Zero
 By now you may have recognized a pattern between the direction of a graphed
 line and its slope:
     If the line is sloping up the right, the slope is positive (+).
    If the line is sloping down to the right, the slope is negative (4.
                                        This line is upward sloping from left to right



             /
                                        so the slope is positive(+). In our pizza
                                        example, a positive slope tells us that as the
                                        number of toppings we order (x) increases,
                                        the total cost of the pizza (y) also increases.
                          X

                                        This line is downward sloping from left to
                                        right so the slope is negative(-). For example,
                                        as the number of people that quit smoking
                                        (x) increases, the number of people
                                        contracting lung cancer (y) decreases. A graph
                                        of this relationship has a negative slope.
                          X
There are two other cases we must consider: (1) when the line on horizontal, and
(2) when the line is vertical.

                                                              change in y
                                                         slope =
                                                              change in x
                                       We can see that no matter what two points
                                       we choose, the value of the y-coordinate stays
                                       the same; it is always 4. Therefore, the change
                                       in y along the line is zero. No matter what
                                       the change in x along the line, the slope must
                                       always equal zero.
                                                                 0
                                                   slope =              =O
                                                            change in x
                                       Horizontal lines have a slope of 0.


22 1 Math Skills for lntroductow Economics
                                                            change in 2
                                                  slope =
                                                            change in x
                                    In this case, no matter what two points we
                                    choose, the value of the x-coordinate stays
                                    the same; its i always 3. Therefore, the
                                                   s
                                    change in x along the line is zero.

      1                                            slope =
                                                           change in y
                                                                .

                                   Since we cannot divide by zero, we say the

                                              -
                                   slope of a vertical line is infinite. The sign for
                                   infinity is .
                                   Vertical lines have an infinite   (0)   slope.


Identifying the Intersection of Two Lines
Now, let's l k k at what happens when there i more than one line on a graph.
                                             s
Many times in the study of economics we have the situation where there is more
than one relationship between the x and y variables. You'll find this type of
occurrence often in your study of supply and demand.
      Y                            In this graph, there are two relationships
                                   between the x and y variables; one
                                   represented by the stra&ht line      and the
                                   other by straight line WZ;We can analyze
                                   each relationship separately or we can look at
                                   them together.


Note in the following graph that RT is downward sloping; it has a negative
slope. Line    is upward sloping; it has a positive slope.




                                                                     Geometry I 23

                                                                                    1934
 In all but one instance, the same y value corresponds to different x values on
 each line. For instance:
         At a y value of 3, the x-value of line   RT is 3 (see point S).
        At a y value of 3, the x value of line                       )
                                                    is 1 (see point D .
 In one case, the two lines have the same (x, y) values simultaneously. This is
 where the two lines RT and        intersect o r cross. The intersection occurs at
 point E which has the coordinates (2,4); the x-coordinate is 2 and the y-coordinate
 is 4.

 Practice #8:
    (a) At what coordinates does the line          intersect the y-axis?
     (b) What are the coordinates of the intersection of lines         and   m?
     (c) At a y value of 8, what is the x value for line   q?
     (dl At point K, what is the y-coordinate?




24 1 Math Skills tot Introductory Economics

                                                                                       03s
           o
 Answers T Practice #8:
    (a)    intersects the y-axis at (0, 10) (Don't forget when you are asked for
        coordinates to give both the x and the y coordinates).
   (b) The coordinates of the intersection of lines G/ and HK are ( 5 3 ) .
   (c) When y has a value of 8, the value of x on line Gf is 2.
   (d) At point K, the y-coordinate is 8.
Again, don't go ahead until you are sure of why all the correct answers are what
they are.




                                                                 Geometry 1 25

                                                                          n        p36
 Nonlinear Relationships

 Most relationships in economics are, unfortunately, not linear. Each unit change
 in the x variable will not always bring about the same change in the y variable.
 The graph of this relationship will be a curve instead of a straight line.
                                      This graph shows a linear relationship
                                      between x and y.




                                         This graph shows a nonlinear relationship
                                         between x and y.




 Determining the Point of Tangency on a Curve
Earlier, we talked about measuring the slope of a straight line. Now we will
discuss how to find the slope of a point on a curve. One of the differences
between the slope of a straight line and the slope of a curve is that the slope of a
straight line is constant, while the slope of a curve changes from point to point.
                          Y




Recall the formula for the slope:




                                         -




26 1 Math Skills for Introductory Economics
 It doesn't matter what pair of points we use to calculate the slope of this line. The
 coordinates for point A are (0,2); the coordinates for B are (1,4). Using these
 points, the slope is:


 Now lefs use point C h D. The coordinates for point C are (2 , ) the coordinates
                                                               6;
 for D are (3, 8). The slope using these points is also


You would get the same slope of 2 with any two points on the line; the slope of a
straight line is constant.
Now let's use the slope formula in a nonlinear relationship.




Let's use our formula for calculating: slope =*chm
                                              ,-
                                                     in y

   From point A (0,2) to point B (1,2!4)       slope=
   From point B (1, 2      to point C (2,4)    slope =      %=*=lx
                                                            2-,

   From point C (2,4) to point D (3,8)         slope= ~ = r = 4 = 4
                                                      8 4

Here we see that the slope of the curve changes as you move along it. For this
reason, we usually measure the slope of a curve at a just one point. For example,
instead of measuring the slope as the change between any t w o points (like A and
B or B and C) we measure the slope of the curve at a single point (like A or C).
To do this we must introduce the concept of a tangent. A tangent i a straight line
                                                                     s
that touches a curve at a single point and does not a o s s through it. The point
where the curve and the tangent meet is called the point of tangency.
Which of these figures illustrates a tangent?




                                                                      Geometry 1 27
        In figures (a) and (b), the straight line is tangent to the curve at point A.
        The straight line just touches the curve at point A but it does not cross
        the curve; the line is tangent to the curve.
        In figure (c), the line and curve are not tangent. The line intersects the
        curve at two points, A and 8.
       In figure (dl, the line touches the curve at a single point but the line
       also crosses the curve, so it is not tangent to the curve.
 The slope of a curve at a point is equal to the slope of the straight line that is
 tangent to the curve at that point.

 Example:
                                   Yl   ID


                                              \
                                              c
                                                    X
The slope of the curve at point B is equal to the slope of the straight line  x.
finding the coordinates of two points on the straight line and u s i n s h e slope
                                                                                   By

                   ,i ,
                   , y
formula (slope = &, n , ) we can determine the slope of the line AC. This will
                    g

also be the slope of t k curve at point 8. Remember, this is the slope of the curve
only at point 8. To find the slope at point D on the curve, we would have to
draw a line tangent to point D and then measure the slope of that tangent.

Practice #5?
   (a) At what point is the straight
       line  tangent to the curve?




    (b) What i the slope of the
             s
        following curve at point B?




28 1 Moth Skills for Introductory Economics
                                                                                        p i g
                                                                                         V
          o
Answers T Practice #9:
   (a) Line  is tangent to the curve at point C.
    (b) The slope of this cwve at point B is   -%.
Remember our formula for the slope of a straight line. Using points A & C, we get:
                        changeiny = 8-0 = 8 --4
                        changeinx 0-6 4 3

Determining Whether the Slope of a Curve is Positive,
Negative, Infinite or Zero
We made some generalizations concerning the slopes of straight lines; we can
also do this with curves.
            Y                                        Y   -




Both (a) and (b) show curves sloping upward from left to right. As with upward
sloping straight lines, we can say that the slope of the curve is positive. While
the slope will differ at each point on the curve, it will always be positive.
To check this, take any point on either curve and draw the tangent of that point.
What-is the slope of the tangent? Positive.

Exampla:
       yI                                A, B, and C are three points on the
                                         curve. Each has a different tangent.
                                         Each tangent has a positive slope,
                                         therefore, the curve has a positive
                                         slope at points A, 8, and C. In fact, any
                                         tangent drawn to the curve will have a
                                         positive slope.
 Both of these next curves are downward sloping. Straight lines that are
 downward sloping have negative slopes; curves that are downward sloping also
 have negative slopes.
              Y                                 Y




 We know, of course, that the slope changes from point to point on a curve, but
 all of the slopes along these two curves will be negative.


                                               A, 8, and C are three points on the
                                              curve. Each has a different tangent.
                                              Each tangent has a negative slope since
                                              it's downward sloping; therefore, the
                                              curve has a negative slope at points A,
                                              8, and C. All tangents to this curve
                                              have negative slopes.

Example:
                                              In this example, our curve has a
                                              positive slope at points A, B, and F, a
                                              negative slope at D, and at points C and
                                              E the slope is zero. (Remember, the
                                              slope of a horizontal line is zero.) Make
                                              sure you understand the logic here
                                              before you move along.
                                   X


                                              (a) At which points is the slope of the
                                                   curve positive?
                                              (b) At which points is the slope of the
                                                   curve negative?
                                              (c) What is the slope at point E?
                                              Take your time. ..we've gotten a little
                                   X          tricky.


30 1 Math Skills for Introductory Economics
                                                                                 , -71
                                                                                          ti4 2
         o
 Answen T Practice # 10:
   (a) The curve is positive only at point D.
   (b) The curve is negative at points A, B, & F.
   (c) The slope of point E is 0 (zero). By the way, the slope is also zero at point C.

Maximum and Maximum Points of Curves
The final two terms we will be dealing with are maximum and minimum
points. As the name implies, the maximum point of a curve is the highest point
on the curve. More technically, it is the point on the curve with the highest y-
coordinate value.
                                           Point A is the highest point on this
                                           curve. It has a greater y-coordinate
                                           value than any other point on the
                                           curve. Point A is maximum point for
                                           the curve.
                             X
                                           There is no maximum point for this
                                           curve. Although we stopped drawing
                                           this curve just past point C, in actuality
                                           the curve keeps going on up so that we
                                           can't say any given point is a
                             X             maximum.

Now we must add another term to our vocabulary. A maximum point is the
point on the curve with the highest y-coordinate and a slope of zero. A
minimum point is the point on the curve with the lowest y-coordinate and a
slope of zero.

Example:
                                          Point C is the maximum point of this
                                          curve. It has the highest y-coordinate
                                          value and the slope of the curve is zero
                                          at that point. How do we know the
                                          slope is zero? Because the tangent to
                                          the curve at C is a horizontal line and
                                          we know that the slope of a horizontal
                                          line is zero.




                                                                      Geometry 1 31

                                                                                        p42
                                              Point A is the minimum point of this
                                              curve. It has the lowest y-coordinate
                                              value and the curve has a slope of zero
                                              at that point.



                                              This curve has no minimum point.
                                              The curve will continue to go down as
                                              we go to the right. Therefore, there is
                                              no point on this curve with the lowest
                                              y-coordinate value and a slope of zero.


Example:




   Some curves have maximums, but no minimums as in figure (a).
   Some curves have minimums, but no maximums as in figure Co).
   Some curves have both maximum and minimum points as in figure (c).
   Some curves have neither maximums nor minimums as in figure (dl.

Practice rir I 1 :
What are the maximum and minimum points (if any) for the following curves?




32 1 Math Skills for Introductory Economics
                                                                                        "('   p43
           o
 Answers T Practice # I I :
    I graph (a):
    n                       In graph (b):        In graph (c):
    Maximum point = C       Maximum point = none Maximum point = B
   Minimum point = B        Minimum point = B    Minimum point = D



 SUMMARY
At this point you should be able to do the following:
   1. Identify different points on a graph using coordinates and the x and y axis
       (pg. 6-8).
   2. Match a graph of a straight line with the appropriate equation (pg. 9-1 1).
  3. Match an equation of a straight line with the appropriate graph (pg. 12-14).
  4. Calculate the slope of a straight line
       (a) from the equation (pg. 15-17).
       (b) from the coordinates of two points (pg. 17-19).
        (c) from the graph (pg. 2Ck23).
    5. Identify whether the slope of a straight line is positive, negative, zero, or
        infinite (pg. 22-23).
    6. Identify the point of intersection between two lines on a graph (pg. 23-25).
    7. Determine points of tangency on a curve and use them to calculate the
        slope of the curve (p. 26-29).
    8. Determine whether a curve (or portion of a curve) has a positive,
        negative, or zero slope (pg. 29-31).
    9. Identify maximum and minimum points of a curve (pg. 31-33).
If you still have questions concerning any of these topics, go back and review
them now. If not, then continue and complete the review test that follows. Try to
d o the test without referring back to the material, but after you have finished the
review test and checked your answers, you are strongly encouraged to reread the
sections on the test questions you have missed.
 Review Examination

 Intructions: For each of the following questions, select the correct answer and
 record it on a scrap p i c e of paper. After you have finish the entire examination
 check your asnwers against the answer key in the back of this book.
    1. The slope of the line (y = a + bx ) is
          (a) y         (b)   a         (c) b              (d) x        (4 %
    2. The slope of the line ( y = K x - 3) is
         (a) )+$       (b) 3            (c) 1           (dl -3          (el   -K
    3. What is the slope of a line that connects point A (12, 6) and point B (7, 1I)?
         (a)           (b)    1         (c) -5          (dl             (e) -1

 Use the following graph for questions 4 through 6 .
                              A




                          0        10   20      30    40      50   60

   4. Which is point on the y-axis?
         (a) A         (b)    B         (c) C        (dl D.

   5. What is the x-coordinate of point B?
         (a) 60        (b)    40        (c) 10          (dl 20           (el 30

   6. Which of the following is an equation of line
      (a) 60 + 2x                     (dl 60 - 2x
         (b) 30 + 2x                            (e) 30 - 2x
         (c) 60 - xX
     -     -      -   -   - --




34 1 Math Skils for Introductocy Economics
    7. What is the slope of line        x?
                                   Y
                                   6
                                   5




       (a) 3         (b) 0              (c) 1      (d)positive    (e) infinite
   8. What is the slope of line        x?
                               A




      (a) 4         (b) -3       (c) 3               (d)   %     (el   -%
Use this graph for questions 9 through 1 1 .
                          Y

                         l01




  9. What is the x-coordinate for point F?
      (a) 6        (b) 8               (c) 0        (dl 4        (el 3
 10. What are the coordinates of the intersection of lines AC and            m?
      (a) (0,3)    (b)    (6,3)        (c) (6,O)    (dl (3,6)    (el (0,6)
   11. At a y value of 4, what is the x value of line          AC   ?
         (a) 0               (b) 6      (c) 1          (d) 7            (el 9

 Use the following graph for questions 12 - 14.




   12. Which of these points is a minimum point of the curve?
        (a) A                (b)   B    (c) C          (d) D            (e) none of these

   13. At which point does the curve have a positive slope?
        (a) A                (b)   B    (c) C          (dl D            (el none of these

   14. .What is the slope at point B?
        (a) 1          (b)   positive   (c) infinite     (dl zero           (e) none of these

  15. Which of these graphs show a line tangent to a curve?




                 (A)                         (B)                           (C)
       (a) B only                               (dl A, B, and C
       (b) A and B only                         (el none of these
       (c) B and C only


             NOTE: Answers To This Review Test Are On The Next Page.


36 1 Math Skills for Introductory Economics
                                                                                            ,   "   fiq 7
Review Examination Answers




If you made any mistakes, we suggest that you review the appropriate
instructional pages.
   1&2      review pages   15-17          8        review pages    19-22
   3        review pages   17-19          9 & 11 - review pages    23-25
   4h5      review pages   6-8            12       review pages    33-33
  6         review pages   9-1 1          13 & 24 review pages     29-31
  7         review pages   19-22          15       review pages    26-29




                                                                Geometry 1 37
                                                                                048
      17.    y = a + bx The slope o f t h i s l i n e is:



 16.        y = 7x 4 ?he slope o f t h i s l i n e is:

             a) 7       b ) -6     C)   13    d)    5       e) x
                                                    7

 1 9 . m i n t A is ( 6 , 1 ) . m i n t R is ( 2 , 5 ) .       bhat   i s t h e slop o f the l i n e that
       c o n m c t s these two p o i n t s .


            Uaz t b follving graph for questions 20-22.




20. ?he p o i n t ( 3 , 2 ) is:

            a) R      b) S       C) T        d) V           e) W

21. 'Ihe o r i g i n is:

            a) R      b) S       C) T        d) V       '   el w

22.     A point     cn the y a x i s is:
            a) S      b) T       c) V        d) w
      23.   mich of the follwing is a graph of the equation y = 20 + x ?   .




              i the equation of thc line shawn in t graph blow?
               s                                   k
        -
24.     h t
  50
                                          a) y = 10 + Sx
  40..                                    b ) y = 10   - 5x
                                          c)y=lO+x
  30-                                     d) y = SO x  '-

                                          el y = 10 + lx
                                                            5
 25.    mt      is the slope o f the! 1 ine ='in              the! following graph?




26.    In the graph below, the s l o p e o f the line                     is:
        a ) p o s i t i v e b ) negative ' c ) a o ( i n f i n i t e ) d ) z e r o e l c a n ' t be d e t e n i n e d




27.    In the gram below, the slope o f t l i b
                                         h                              is:
        a) 0           b) 1            c) 2             dl-     (infinite)
                              e)     1000
         Use this graph for questions 28-32.




                                      QUANTITY
28.    A t point A, w h t is the price?




29.   A t point A, what is the quantity?

         a ) 200    b ) 300        c ) 400     d ) SO0    el 600

30.   The s q p l y line intersects the y a x i s a t a price equal to;

        a ) $1      b) $2          c ) $3      d ) $4    el $5

31.   A t what quantity      do the s u p ~ l y d demand lines intersect?
                                               a
        a)   sz     b ) SI         C)WO      d)$1        el NO

32.   A t a price of $1, utmt i s the e ~ a n t i t ydemanded?

        a1300      b) 400          c ) 500    dl600      el700
        U8a t h i s graph for questions 33-38.




33.    A t which point is t h e slope o the cum.negative?
                                       f
         a) P        b) 0         C)    R    dl S     el T
34.    A t which point is the slope o the curve p i t i v e ?
                                     f
         a) R       b) S     '   C )   T    d) V
35.   A t which point is tb slope o the curve zero?
                                   f


36.                                     u
      bhich point lies on a part o the c m that has a decreasingly
                                  f
      p a s i t i v e slop?


37.       *
      A t .t    point is t straight l i n e tangent to the curve?
                          b
        a) P        b) S         c) V       dl W     e) T
38.   Which point is a m a x i m point for tk curve?
        a) P       b) R          c) S       dl T     e ) v-
Mathematics Skill Assessment For Economics
               (Answer Key)

								
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