Pre-Algebra Chapter 8 Functions and Graphing 8-1 Relations and Functions Goals of Chapter 8 Determine whether relations are functions. Graph data in scatter plots. Graph linear equations using slope and intercepts. Solve systems of equations. Graph inequalities. Modeling a Real-World Application: Ecology page 372 – Have you ever thought you couldn’t make a difference in landfills by recycling? Think again! The table below shows how the United States compares to other nations in terms of the amount of annual household waste. These five countries produce the most waste in the world.
Equivalent Annual Domestic Waste per Person (thousands of tons) (pounds) 15,500 634 15,816 620 40,225 634 200,000 1925 20,780 741
Country France Great Britain Japan United States West Germany
This data could also be displayed using a set of ordered pairs. Each first coordinate would be the annual domestic waste, and each second coordinate would be the equivalent per person. {(15,500, 634), (15,816, 620), (40,225, 634), (200,000, 1925), (20,780, 741)} A relation is a set of ordered pairs, like the set above. The set of first coordinates is called the domain of the relation. The set of second coordinates is called the range of the relation. The domain of the above relation is {15,500, 40,225, 200,000, 20,780}, and the range is {634, 620, 1925, 741} Notice that 634 is listed only once. A relation can also be modeled by a table or a graph.
x -3 0 4 y 1 -2 -2
.
.
Example 1 page 373 – Express the relation shown in the graph as a set of ordered pairs. Then determine the domain and range. A function is a relation in which each element of the domain is paired with exactly one element in the range. Example 2 page 373 – Graph each relation. Then determine whether each relation is a function. a) b)
x y 6 0 3 8 6 -1 -2 3 x y -3 1 -2 -2 0 1 2 3
You can use the vertical line test to determine if a relation is a function. For each value of x, any vertical lie passes through no more than one point on the graph. This is true for every function. Assign 8-1 page 375 # 17 – 43 odd, 44 – 51
Pre-Algebra Chapter 8 Functions and Graphing 8-2 Scatter Plots First do the Hands-on Activity on page 378. Modeling with Technology – page 379 The sales and advertising costs for ten of the largest U.S. industrial corporations are shown in the chart below. Use a graphing calculator to create a scatter plot of the data. A scatter plot is a graph that show the general relationships between two sets of data.
Sales Advertising Costs (millions of dollars) (millions of dollars) Company General Motors 132,775 133.6 Ford 100,786 794.5 IBM 65,096 185.5 General Electric 62,202 250.7 Philip Morris 50,157 2024.1 Chrysler 36,897 756.6 Procter & Gamble 29,890 2165.6 PepsiCo 22,084 928.6 Eastman Kodak 20,577 686.6 Dow Chemical 19,177 186.6
Before you create a scatter plot, clear the statistical memories. Page 379 Steps to enter the scatter plot on page 379. If the points in a scatter plot appear to suggest a line that slants upward to the right there is a positive relationship. If the points in a scatter plot appear to suggest a line that slants downward to the right there is a negative relationship. If the points in a scatter plot seem to be random, then there is no relationship. Assign 8-2 page 382 #11 – 23 odd, 24, 25, 27, 29 – 33
Pre-Algebra Chapter 8 Functions and Graphing 8-3 Graphing Linear Relations Modeling Real-World Application: Science page 385 – Did you know that a cricket is nature’s thermometer? You can use the equation t = c + 40 to find the temperature t if you know the number of chirps c a cricket makes in 15 seconds.
Chirps in 15 seconds 21 23 31 50 47 c + 40 Temperature (F) 21 + 40 61 23 + 40 63 31 + 40 71 50 + 40 90 47 + 40 87
Graphing Linear Relations Recall that solving an equation means to replace the variable so that a true sentence results. The solutions of an equation with two variables are ordered pairs. To find a solution of such an equation, choose any value for x, substitute that value into the equation, and find the corresponding value for y. It is often convenient to organize the solutions in a table. Example 1 page 385 – a) Find four solutions for the equation y = -2x + 1. Write the solutions as ordered pairs. b) Graph the equation y = -2x + 1. An equation like y = -2x + 1 is called a linear equation because its graph is a straight line. An equation with two variables has an infinite number of solutions. Example 2 page 386 – In any right triangle, the sum of the measures of the two acute angles is 90. If x and y represent the measures of the two acute angles, then the equation x + y = 90 models this condition. a) Graph the equation. b) Use the graph to name another solution of the equation. c) Is (-10, 100) a solution of the equation? Assign 8-3 page 388 # 21 – 55 odd, 56, 57, 59 – 70
Pre-Algebra Chapter 8 Functions and Graphing 8-4 Equations as Functions Modeling with Manipulatives – page 392 Use the integers -5 through 4 to find the output for each number. Input Rule Output
x2 -3
The equation in the above function involved using two variables. Since the solutions of an equation in two variables are ordered pairs, such an equation describes a relation. The set of values of x is the domain of the relation. The set of corresponding values of y is the range. Example 1 page 392 – a) Solve y = 3x + 4 if the domain is {-2, -1, 0, 1} b) Graph the equation. c) Is the equation a function? Equations that represent functions can be written in functional notation, f(x). The symbol f(x) is read “f of x” and represents the value in the range that corresponds to the value of x in the domain. Example 2 page 393 – If g(x) = 3x + 4, find each of the following. a) g(3) b) g(-5) c)2 g(0) Example 3 page 393 – Oceanographers use sonar to locate fish and other objects beneath the surface of water. Sonar units emit a pulse or sound and calculate the distance to the object using the time it takes for the pulse to reflect back. You can use the function vt = d, where v is the velocity, t is the time, and d is the distance to find the time. The speed or velocity of sound in water is 1454 meters per second. How far below the surface is a school of fish if it takes 0.022 seconds for the sound to return to the sonar? The sound must go to the fish and return, so the time to the fish is 0.022 2 or 0.011 seconds. vt = d (1454)(0.011) = d 15.994 = d The fish are about 16 meters from the sonar. Assign 8-4 page 394 # 13 – 31 odd, 32, 33, 35 – 40
Pre-Algebra Chapter 8 Functions and Graphing 8-5 Problem-Solving Strategy Draw a Graph Modeling a Real-World Application : Air Travel page 396 – Imagine you are siting in your summer clothes and sandals eating lunch and it’s 74 degrees below zero! It happens every day on high-flying airplanes that are far from the heat of Earth’s surface. The temperature at different altitudes above sea level when the temperature at sea level is 60F. The table does not include an altitude for -74F. At what altitude would a jet be flying when the thermometer has this reading?
Altitude (thousands of feet) Temp (?F) 5 41 10 23 15 4 20 -15 25 -33
Use a graph is represent the same information as its equation. But, because a graph is visual, it allows you to see patterns that may not be obvious from the equation. A graph is a powerful tool in problem solving. You can use a graphing calculator to plot points and construct a line without erasing the points. Then you can use the trace function to answer the question. Example 2 page 397 – The ordered pairs (50.7, 16.47) and (99.92, 25.42) represent data from two rows of a table comparing cubic meters of natural gas used and the cost. What would the bill for 140 cubic meters of natural gas be? Assign 8-5 page 398 # 6 – 17 Self Test page 399 # 1 – 10
2 days Pre-Algebra Chapter 8 Functions and Graphing 8-6 Slope Modeling a Real-World Application : Construction page 400 – In many buildings, the wheelchair ramps are built next to the walls with stairways in the middle. If you were to compare the handrail on the ramp to the handrail on the stairway, you would notice that the handrail on the ramp is not as steep as the one for the stairs. The steepness of the handrails depends on the vertical change and the horizontal change. It can be expressed as a ratio. verticalchange steepness horizontalchange In mathematics, the slope m of a line describes its steepness. The vertical change is called the change in y, and the horizontal change is called the change in x.
slope
changeiny changeinx
Example 1 page 400 – Find the slope of the line graphed in the book. Example 2 page 401 – Find the slope of the line that contains A(-2, 5) and B(4, -5). Then graph the line. Sometimes the vertical change is referred to as the rise, and the horizontal change is referred to as the run. You can remember slope as rise over run.
slope
rise run
Example 3 page 401 – Signs indicated the slope of an upcoming hill are often posted in hilly areas. Find the slope of a road that increases 633.6 feet in 2 miles. The graph of two lines that never intersect are called parallel lines. These lines have a special relationship between the slopes. What is it?
Assign 8-6 page 403 # 15 – 33 odd, 34 – 44
2 days Pre-Algebra Chapter 8 Functions and Graphing 8-7 Intercepts Modeling a Real-World Application : Sports page 406 – On your mark, get set, go! Sports analysts have found that the world record for the 10,000 meter run has been decreasing steadily since 1940. The trend ca be described by the equation y = 30.18 – 0.07x, where x is the number of years since 1940 and y is the record time. the graph of the equation is a line that crosses the x-axis at approximately (431,0) and crosses the y-axis at (0,30.18). The points where a graph crosses the axes are called intercepts. The graph of a linear function may cross either the x-axis, the y-axis, or both axes. The x-intercept is the x-coordinate of the point where the graph crosses the x-axis. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. Think – What kind of line only crosses the x-axis? What kind of line only crosses the y-axis? Example 1 page 407 – Graph
y
3 x 6 using the x- and y-intercepts. 2
To find the y-intercept, let the x = 0 (0, )
To find the x-intercept, let y = 0. ( ,0)
How can you find the y-intercept from the equation? Slope-intercept form y = mx +b Slope y-intercept You can use the slope-intercept form of an equation to graph a line quickly. Example 2 page 407 – The cost for a child to attend The Learning Station is $19 a day plus a registration fee of $30. Make a graph that can be used to find the cost of any length of attendance. C=19d+30 Assign 8-7 page 409 # 17 – 45 odd, 46 – 53
Pre-Algebra Chapter 8 Functions and Graphing 8-8 Systems of Equations Modeling with Technology page 412 – Between 1984 and 1994, the number of cellular phone users increased more than 7 times. In a questionnaire concerning phone service, potential customers were asked whether they would prefer to pay $10 a month and 10 cents for each local call or $5 a month and 20 cents for each local call. Is there a certain number of calls where the two payment plans result in the same total? To solve this problem, let n represent the number of local calls made in a month and let c represent the monthly cost. Plan 1 c = 10 + 0.1n Plan 2 c = 5 + 0.2n
You can use a graphing calculator to solve the problem quickly. Enter each equation into the calculator and use the table feature to find the value of n for which the value of n is the same. Two equations together are called a system of equations. The solution to this system is the ordered pair that is a solution of both equations. Another method for solving a system of equations is to graph the equations on the same coordinate plane. The coordinates of the point where the graphs intersect is the solution of the system of equations. Example 1 page 413 – Solve the system of equations y = 2x and y = -x + 3 by graphing. Given 2 lines the three possible solutions: 1. One point (intersecting lines). 2. No solution (parallel lines). 3. Infinitely many solutions (same line). Example 3 page 413 – Solve each system of equations by graphing. a) y = x + 3 and y = x + 4 (parallel lines) b) y = -2x + 3 and 3y = -6x + 9 (same line) Assign 8-8 page 415 # 15 – 39 odd, 40 – 46
2 days Pre-Algebra Chapter 8 Functions and Graphing 8-9 Relations and Functions Modeling with Manipulatives page 418 – The graph of y = x + 2 is shown in the book. Copy the axes and the graph on a piece of graph paper. Graph the following ordered pairs on your coordinate system. (3,4) (3,-2) (-2, -1) (-1, 0) (4, 4) (0, -3) a) Where do these points lie in the plane in relation to the graph of y = x + 2? b) Which sentence y = x + 2, y > x + 2, or y < x + 2, is true for all of the ordered pairs you graphed? c) Which sentence, y = x + 2, y > x + 2, or y < x + 2, is true for the points located above the graph of y = x + 2? d) How many points belong to the graph of y < x + 2? To graph an inequality: 1) First draw the corresponding equation. The line is the boundary of the two regions. 2) Decide if the boundary line is solid or dashed. 3) Test a point to see if the inequality is true. 4) If the inequality is true, shade in the direction of that point. If the inequality is false, shade in the opposite direction of that point. Example 1 page 419 - Graph y < -2x + 2. Example 2 page 420 – Tickets for West Side Story are $66.50 for floor and $47.50 for balcony. In order to cover the expenses for the show, at least $66,500 per show must be made from ticket sales. a) Write an inequality to represent the situation. b) Use a graph to determine how many of each type of ticket must be sold to cover the expenses. c) List three possible solutions.
Assign 8-9 page 421 # 19 – 43 odd, 44, 45, 47, 49 – 56 Study Guide and Assessment page 423
Pre-Algebra Quiz after Lesson 8-2 1) What is a relation? 1)
Name (2)
Use the data below to answer the questions 2 & 3.
Chirps in 15 seconds Temperature in degrees F 5 45 10 50 15 55 20 60
2) Write the domain of the above relation. 2) 3) Write the range of the above relation. 3) 4)
(2) (2) (1)
Determine whether each relation is a function. 4)
x y 0 2 2 4 4 2 6 4
5)
x y 0 4 3 5 4 6 3 7
5)
(1)
6) Name the 5 steps you go through to do a scatter plot on the graphing calculator. 1) (1) 2) (1) 3) (1) 4) (1) 5) (1) Use the table below to answer question 7.
Sales Advertising Costs (millions of dollars) (millions of dollars) Company General Motors 132,775 133.6 Ford 100,786 794.5 IBM 65,096 185.5 General Electric 62,202 250.7 Philip Morris 50,157 2024.1 Chrysler 36,897 756.6 Procter & Gamble 29,890 2165.6 PepsiCo 22,084 928.6 Eastman Kodak 20,577 686.6 Dow Chemical 19,177 186.6
7) If you were to graph the above data, what would your Window settings look like: WINDOW Xmin = (1) Xmax = (1) Xscl = (1) Ymin = (1) Ymax = (1) Xscl = (1) Xres = (1) 8) On the back, explain what it means for a relation to have a positive relationship, negative relationship, and no relationship. (5 - extra points for a great answer)