Finding the Equation from a Slope and y-intercept

Transcription

1 Lesson 4.4 Objectives Write linear equations given a slope and y-intercept, a slope and a point, or a graph. Writing Linear Equations Michael turns on the high-temperature oven each morning when he comes to the quality assurance laboratory. A chart recorder shows the evenly rising temperature over time. Michael knows it takes almost an hour for the oven to reach the desired temperature of 1,200F. But, how long will it take to reach 800F? Finding the Equation from a Slope and y-intercept The most common way to write a linear equation is in slopeintercept form. This is the most common way since it provides an easy way to identify both the slope and y-intercept of the graph. When you are given the slope of a line, m, and the y-intercept, b, you can substitute these values into the slope-intercept form to write a linear equation. Example 1 Finding the Equation of a Line Write the equation of the line that has a slope of 2 and y-intercept (0, 7) in slope-intercept form. Solution Substitute 2 for m and 7 for b in slope-intercept form. y mx b y 2 x Writing Linear Equations 22

2 Finding the Equation from a Point and a Slope In Lesson 4. you graphed a line given a point and the slope. You can write the equation of the line without graphing using either the slope-intercept form y mx b or the point-slope form. To use the slope-intercept form, use the coordinates of the given point to calculate b. To use the point-slope form, use the slope and the given point in the formula below. Point-Slope Form of a Linear Equation y y 1 m(x x 1 ), where (x 1, y 1 ) is a point on the line and m is the slope of the line. Example 2 Finding the Equation of a Line The phone company charges Jerry $0.45 for each additional peak minute over 500 minutes for using his cell phone. His bill last month was $60.25 for 545 minutes. What linear equation models Jerry s monthly cell phone bill? Solution Let x represent the number of additional peak minutes used and let y represent the total cost of the phone bill. The ordered pair (45 minutes, $60.25) represents a point on the graph of the equation. In slope-intercept form, the equation will be y mx b. The slope m is the charge rate. So m $0.45 per minute. Use the equation y 0.45x b. To find the y-intercept b, use the point (45, 60.25) that is on the line (45) b b 40 b Using the equation y $0.45x $40, you can determine the amount of Jerry s monthly bill for x additional peak minutes. Ongoing Assessment a. What is Jerry s monthly cell phone bill if he uses 62 additional peak minutes? b. How many additional peak minutes did Jerry use if his monthly bill was $4.15? 224 Chapter 4 Linear Equations

3 Ongoing Assessment Write the equation of the line that has the slope given and passes through the given point. a. (4, 5) and slope 2 y 2x 1 b. (4, 1) and slope 7 y x x c. (1, 6) and slope 1 4 y 1 4 x 41 4 Finding the Equation from a Graph Another way you can find the equation of a line is by analyzing its graph. The approach is very similar to the previous examples. In some cases you can simply look at the graph and determine the slope and y-intercept. In other cases, this may be more difficult to determine just by looking at the graph. Example Write an Equation from a Graph Reread the opening paragraph of this lesson. After Michael turns on the oven, how long will it take to reach 800F? Solution Model the relation between time (x) and oven temperature (y) with a linear equation. Oven Temperature Temperature ( F) Time (minutes) At 10 minutes he reads a temperature of 250F. This represents the ordered pair (10, 250). At 20 minutes he reads a temperature of 400F. This represents the ordered pair (20, 400). 4.4 Writing Linear Equations 225

4 Michael uses these two ordered pairs to find the slope. m y 2 y1 x2 x To determine the equation of the line, he needs to find the y-intercept. Use the point on the line (10, 250). point-slope form y-intercept form y y 1 m(x x 1 ) y mx b y (x 10) (10) b y x b y x b y 15x 100 b 100 y 15x 100 The equation that models the oven temperature is y 15x 100. Critical Thinking Use the graph to find the number of minutes it will take (x) for the oven to reach 800F (y). What temperature does the oven reach in 0 minutes? Ongoing Assessment Select two points on the graph and determine the equation for the line. Predict the value for another point on the line. 226 Chapter 4 Linear Equations

5 Lesson Assessment Think and Discuss 1. Explain how to find the equation of a line if you know the slope and y-intercept. 2. If you are given a point on a line and the slope of the line, what equation can you use to determine the y-intercept of the line?. Explain how to find the slope of a graph if you can identify two points on the graph. 4. What will be the form of the equation of a line that has a slope of 0? Practice and Problem Solving Write an equation in slope-intercept form for the line defined by the given slope th and intercept. 5. slope 1 ; y-intercept (0, 0) 2 6. slope 2; y-intercept (0, ) 7. slope 2 ; y-intercept (0, 5) 8. slope 4 ; y-intercept (0, 9) 5 9. slope 0; y-intercept S0, 2D 10. slope 4 ; y-intercept (0, 6) 11. slope 5; y-intercept (0, 7) 12. slope undefined; x-intercept (2, 0) Write an equation in slope-intercept form for the line defined by the given slope the and given point on the line. 1. slope 1 ; (4, 2) 14. slope ; (0, 7) slope 2 ; (9, 4) 16. slope 0; (5, 10) 17. slope 7 ; (5, 2) 18. slope 5 ; (6, 12) slope 1 ; (, 4) 20. slope 1; (4, 0) Writing Linear Equations 227

6 Write an equation in slope-intercept form for each line graphed. Identify the y-intercept Chapter 4 Linear Equations

7 27. Helen manages an apartment building. Upstairs tenants often want to install waterbeds, but Helen must determine whether the floor can support the weight of the beds. The typical frame weighs 80 pounds, and she knows that water weighs 8. pounds per gallon. a. What is the slope of the line that represents this situation? b. What is the y-intercept of the line that represents this situation? c. Write the equation of the line in slope-intercept form. d. Each square foot of a waterbed contains approximately 4 gallons of water. If a bed is 5 feet wide and 7 feet long, what will be the total weight of the waterbed? Mixed Review Write each number in scientific notation ( )( ) 0. A carpenter is cutting circular blocks from plywood. If the diameter of the circle is 2.1 meters, what is the area of the plywood section being cut out? 1. Tim invested $5,250 at 5.5% simple interest for 18 months. How much interest did Tim receive? 2. Draw a circle so that it fits into a square exactly. If the length of one side of the square is 625 centimeters, what is the area of the circle? Evaluate each expression when x is 4 and y is 2.. 5x y 4. x(y 5) 5. x 2 y 4.4 Writing Linear Equations 229