4.1 Write Linear Equations in slope-intercept form Essential Question: You will write equations of lines. How do you write an equation of a line in slope-intercept form? You will learn how to answer this question by using the slope and y-intercept or two points to write an equation of the line.

Use slope and y-intercept to write an equation EXAMPLE 1 Use slope and y-intercept to write an equation Write an equation of the line with a slope of –2 and a y-intercept of 5. y = mx + b Write slope-intercept form. y = –2x + 5 Substitute –2 for m and 5 for b.

Standardized Test Practice EXAMPLE 2 Standardized Test Practice Which equation represents the line shown? A y = – x + 3 2 5 B C y = – x + 1 D y = 3x + = – The slope of the line is rise run –2 5 2 . The line crosses the y-axis at (0, 3). So, the y-intercept is 3. y = mx + b Write slope-intercept form. 2 y = – x + 3 5 2 Substitute – for m and 3 for b. 5

EXAMPLE 2 Standardized Test Practice ANSWER The correct answer is A. B D C A

GUIDED PRACTICE for Examples 1 and 2 Write an equation of the line with the given slope and y-intercept. 1. Slope is 8; y-intercept is –7. y = 8x – 7 ANSWER

GUIDED PRACTICE for Examples 1 and 2 Write an equation of the line with the given slope and y-intercept. 2. Slope is ; y intercept is –3. 3 4 y = 3 4 x – 3 ANSWER

EXAMPLE 3 Write an equation of a line given two points Write an equation of the line shown.

Write an equation of a line given two points EXAMPLE 3 Write an equation of a line given two points SOLUTION STEP 1 Calculate the slope. x2 – x1 3 3 – 0 y2 – y1 = m –1 – (–5) 4 STEP 2 Write an equation of the line. The line crosses the y-axis at (0, –5). So, the y-intercept is –5. y = mx + b Write slope-intercept form. y = x – 5 4 3 Substitute for m and 5 for b. 4 3

EXAMPLE 4 Write a linear function Write an equation for the linear function f with the values f(0) = 5 and f(4) = 17. SOLUTION STEP 1 Write f(0) = 5 as (0, 5) and f (4) = 17 as (4, 17). STEP 2 Calculate the slope of the line that passes through (0, 5) and (4, 17). x2 – x1 4 – 0 y2 – y1 = m 17 – 5 4 12 3

Write a linear function EXAMPLE 4 Write a linear function STEP 3 Write an equation of the line. The line crosses the y-axis at (0, 5). So, the y-intercept is 5. y = mx + b Write slope-intercept form. y = 3x + 5 Substitute 3 for m and 5 for b. ANSWER The function is f(x) = 3x + 5. • How does f (4) = 17 correspond to the ordered pair (x, y )?

GUIDED PRACTICE for Examples 3 and 4 3. Write an equation of the line shown. 2 1 x + 1 y = ANSWER –

GUIDED PRACTICE for Examples 3 and 4 Write an equation for the linear function f with the given values. 4. f(0) = –2, f(8) = 4 y = x – 2 3 4 ANSWER

GUIDED PRACTICE for Examples 3 and 4 Write an equation for the linear function f with the given values. 5. f(–3) = 6, f(0) = 5 y = x + 5 1 3 ANSWER –

EXAMPLE 5 Solve a multi-step problem Recording Studio A recording studio charges musicians an initial fee of $50 to record an album. Studio time costs an additional $35 per hour. Write an equation that gives the total cost of an album as a function of studio time (in hours). a. Find the total cost of recording an album that takes 10 hours of studio time. b.

EXAMPLE 5 Solve a multi-step problem SOLUTION The cost changes at a constant rate, so you can write an equation in slope-intercept form to model the total cost. a.

EXAMPLE 5 Solve a multi-step problem STEP 1 Identify the rate of change and the starting value. Rate of change, m: cost per hour Starting value, b: initial fee STEP 2 Write a verbal model. Then write the equation. C = t 35 + 50

Solve a multi-step problem EXAMPLE 5 Solve a multi-step problem Use unit analysis to check the equation. CHECK hour dollars = hours + dollars dollars The total cost C is given by the function C = 35t + 50 where t is the studio time (in hours). ANSWER b. Evaluate the function for t = 10. C = 35(10) + 50 = 400 Substitute 10 for t and simplify. ANSWER The total cost for 10 hours of studio time is $400.

GUIDED PRACTICE for Example 5 WHAT IF? In Example 5, suppose the recording studio raises its initial fee to $75 and charges $40 per hour for studio time. 6. Write an equation that gives the total cost of an album as a function of studio time (in hours). a. C = 40t +75 ANSWER

GUIDED PRACTICE for Example 5 Find the total cost of recording an album that takes 10 hours of studio time. b. $475 ANSWER

Daily Homework Quiz Write an equation of the line with a slope of –4 and a y-intercept of 1. 1. ANSWER y = –4x + 1 Write an equation of the line that passes through the given points. (–9, 1), (0, –8) 2. ANSWER y = –x – 8 (–4, –6), (0, 6) 3. ANSWER y = 3x + 6

Daily Homework Quiz Write an equation for the linear function f with f(0) = –4 and f(–1) = –9. 4. ANSWER y = 5x – 4 An electronics game store sells used games for $12.99 with a $20 membership fee. Write an equation that gives the total cost to become a member and buy games as a function of the number of games that are purchased. Then find the cost for 6 games. 5. ANSWER C = 12.99g + 20 where C is total cost and g is the number of games; $97.94

Essential Question: You will write equations of lines. How do you write an equation of a line in slope-intercept form? • An equation in slope-intercept form, y = mx+ b, has slope m and y-intercept b. • Given two points on a graph, one of which gives the y-intercept, you can find the slope m and the y-intercept b and substitute those values into y = mx + b. • You can write a function in slope-intercept form given values of f (0), x, and f (x). To write an equation in slope intercept form, substitute the slope for m and the y-intercept for b in the equation y = mx + b. In real-world situations, use the constant rate of change for m and the starting value for b.