Warm-Up

1. 5.5: Students will be able to use the point-slope form to write an equation of a line. Warm-Up Write an equation of the line for each problem (1-3). 1. passes through (3, 4), m = 3 ANSWER y = 3x– 5 2.passes through (–2, 2) and (1, 8) ANSWER y = 2x + 6 3.A carnival charges an entrance fee and a ticket fee. One person paid $27.50 and brought 5 tickets. Another paid $45.00 and brought 12 tickets. How much will 22 tickets cost? ANSWER $70

2. Review Homework

3. 3. A camp charges a registration fee and a daily amount. If the total bill for one camper was $338 for 12 days and the total bill for another camper was $506 for 19days what will the total bill be for a camper who enrollsfor 30 days? ANSWER ANSWER ANSWER $770 y = –2x –3 y = 2x – 1 (2,3), (4,7) (–5,7), (2, –7) 1. 2. Daily Homework Quiz Write an equation of the line that passes through the given point.

4. Vocabulary Methods to Represent Linear Functions Slope Intercept Form: y = mx + b Point-Slope Form: y – y1 = m(x – x1) m = slope (x1, y1) = point on the line

5. EXAMPLE 1 Write an equation in point-slope form Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2. Write point-slope form. y –y1= m (x –x1) y +3=2 (x–4) Substitute2form,4 for x1, and–3 fory1.

6. 1. Write an equation in point-slope form of the line that passes through the point (– 1, 4) and has a slope of –2 . EXAMPLE 1 for Example 1 Write an equation in point-slope form GUIDED PRACTICE y –y1= m (x –x1) Write point-slope form. y –4=–2 (x +1) Substitute –2 for m,4 fory, and –1 forx.

7. 2 3 y+2 = (x– 3). EXAMPLE 2 Graph an equation in point-slope form Graph the equation. Hint y-y1 = m(x-x1)

8. SOLUTION Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2). 2 3 EXAMPLE 2 Graph an equation in point-slope form Plot the point (3, – 2). Find a second point on the line using the slope. Draw a line through both points.

9. Graph the equation. 2. – y–1 = (x– 2) EXAMPLE 2 for Example 2 Graph an equation in point-slope form GUIDED PRACTICE SOLUTION Because the equation is in point-slope form, you know that the line has a slope of –1 and passes through the point (2, 1). Plot the point (2, 1). Find a second point on the line using the slope. Draw a line through both points.

10. EXAMPLE 3 Use point-slope form to write an equation Write an equation in point-slope form of the line shown.

11. 3 –1 y1 – y2 2 = = m = = –1 – 1 –1 x1 – x2 – 2 EXAMPLE 3 Use point-slope form to write an equation SOLUTION STEP1 Find the slope of the line.

12. EXAMPLE 3 Use point-slope form to write an equation STEP 2 Write the equation in point-slope form. You can us either given point. Method 1 Method 2 Use (– 1, 3). Use (1, 1). y –y1= m(x –x1) y –y1=m(x – x1) y –3=–(x +1) y –1=–(x – 1) CHECK Check that the equations are equivalent by writing them in slope-intercept form. y – 3 = –x – 1 y – 1 = –x + 1 y = –x + 2 y = –x + 2

13. 3. Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). 1 = 2 4 –3 y2 – y1 = = m 4 –2 x2 – x4 EXAMPLE 3 for Example 3 Use point-slope form to write an equation GUIDED PRACTICE STEP1 Find the slope of the line.

14. 1 1 2 2 y –3=(x – 2) y –4= (x – 4) EXAMPLE 3 for Example 3 Use point-slope form to write an equation GUIDED PRACTICE STEP 2 Write the equation in point-slope form. You can us either given point. Method 1 Method 2 Use (2, 3) Use (4, 4) y –y1= m(x –x1) y –y1=m(x – x1)

15. EXAMPLE 4 Solve a multi-step problem Stickers You are designing a sticker to advertise your band. A company charges $225 for the first 1000 stickers and $80 for each additional 1000 stickers. Write an equation that gives the total cost (in dollars) of stickers as a function of the number (in thousands) of stickers ordered. Find the cost of 9000 stickers.

16. EXAMPLE 4 Solve a multi-step problem SOLUTION STEP1 Identify the rate of change and a data pair. Let Cbe the cost (in dollars) and s be the number of stickers (in thousands). Rate of change, m: $80 per 1 thousand stickers Data pair(s1, C1): (1thousand stickers,$225)

17. EXAMPLE 4 Solve a multi-step problem STEP2 Write an equation using point-slope form. Rewrite the equation in slope-intercept form so that cost is a function of the number of stickers. C –C1=m(s –s1) Write point-slope form. C–225=80(s–1) Substitute 80 for m, 1 for s1, and 225 for C1. C = 80s +145 Solve for C.

18. ANSWER The cost of 9000 stickers is $865. EXAMPLE 4 Solve a multi-step problem STEP3 Find the cost of 9000 stickers. Substitute 9 for s. Simplify. C = 80(9) + 145 = 865

19. Number of people 4 6 8 10 12 Cost (dollars) 650 250 350 450 550 EXAMPLE 5 Write a real-world linear model from a table Working Ranch The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group.

20. 650 – 550 = 50 12 – 10 350 – 250 450 – 350 = 50, = 50, 550 – 450 8 – 6 6 – 4 = 50, 10 – 8 EXAMPLE 5 Write a real-world linear model from a table SOLUTION STEP1 Find the rate of change for consecutive data pairs in the table. Because the cost increases at a constant rate of $50 per person, the situation can be modeled by a linear equation.

21. EXAMPLE 5 Write a real-world linear model from a table STEP2 Use point-slope form to write the equation. Let Cbe the cost (in dollars) and pbe the number of people. Use the data pair (4, 250). C –C1=m(p –p1) Write point-slope form. C–250=50(p– 4) Substitute 50 for m, 4 for p1, and 250 for C1. C = 50p +50 Solve for C.

22. for Examples 4 and 5 GUIDED PRACTICE 4. WHAT IF? In Example 4, suppose a second company charges $250 for the first 1000 stickers. The cost of each additional 1000 stickers is $60. a. Write an equation that gives the total cost (in dollars) of the stickers as a function of the number(in thousands) of stickers ordered. b. Which Company would charge you less for 9000 stickers?

23. for Examples 4 and 5 GUIDED PRACTICE SOLUTION STEP1 Identify the rate of change and a data pair. Let Cbe the cost (in dollars) and sbe the number of stickers (in thousands). Rate of change, m: $60 per 1 thousand stickers Data pair(s1, C1): (1thousand stickers,$250)

24. for Examples 4 and 5 GUIDED PRACTICE STEP2 Write an equation using point-slope form. Rewrite the equation in slope-intercept form so that cost is a function of the number of stickers. C –C1=m(s –s1) Write point-slope form. C–250=60(s– 1) Substitute 250 for C1, 60 for m, and 1 for s1. C = 60s +190 Solve for C.

25. ANSWER The cost of 9000 stickers is $730. The second Company would charge you less for 9000 stickers. for Examples 4 and 5 GUIDED PRACTICE STEP3 Find the cost of 9000 stickers. Substitute 9 for s. Simplify. C = 60(9) + 190 = 730

26. for Examples 4 and 5 GUIDED PRACTICE Mailing Costs The table shows the cost (in dollars) of sending a single piece of first class mail for different weights. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost of sending a piece of mail as a function of its weight (in ounces). Weight (ounces) 1 4 5 10 12 Cost (dollars) 2.90 0.37 1.06 1.29 2.44

27. 2.44 – 1.29 1.06 – 0.37 1.29 – 1.06 = 0.23, = 0.23, = 0.23, 10 – 5 4 – 1 5 – 4 2.90 – 2.44 = 0.23 12 – 10 for Examples 4 and 5 GUIDED PRACTICE SOLUTION STEP1 Find the rate of change for consecutive data pairs in the table. Because the cost increases at a constant rate of $0.23 per mail, the situation can be modeled by a linear equation.

28. for Examples 4 and 5 GUIDED PRACTICE STEP2 Use point-slope form to write the equation. Let Cbe the cost (in dollars) and wbe the weight of mails. C –C1=m(w –w1) Write point-slope form. C–0.37=0.23(w– 1) Substitute 0.23 for m, 1 for w1, and 0.37 for C1. C–0.37=0.23w– 0.23 Simplify C = 0.23w +0.14 Solve for C.