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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)