Warm-Up

Warm-Up Exercises5.5: Students will be able to use the point-slope form to write an equation of a line.

Write an equation of the line for each problem (1-3).

2. passes through (–2, 2) and (1, 8)

ANSWER

ANSWER

1. passes through (3, 4), m = 3

y = 2x + 6

y = 3x – 5

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

Warm-Up

Warm-Up ExercisesReview Homework

Warm-Up Exercises

ANSWER $770

ANSWER y = –2x – 3

ANSWER y = 2x – 1

Daily Homework Quiz

(2, 3), (4,7)1.

(–5, 7), (2, –7)2.

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 19 days what will the total bill be for a camper who enrolls for 30 days?

3.

Write an equation of the line that passes through the given point.

Methods to Represent Linear Functions

Slope Intercept Form: y = mx + bPoint-Slope Form: y – y1 = m(x – x1)

m = slope(x1, y1) = point on the line

Vocabulary

Warm-Up ExercisesWrite an equation in point-slope form

EXAMPLE 1

Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2.

y – y1 = m (x – x1) Write point-slope form.y + 3 = 2 (x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.

Warm-Up ExercisesWrite an equation in point-slope form

EXAMPLE 1

y – y1 = m (x – x1) Write point-slope form.

y – 4 = –2 (x +1) Substitute – 2 for m, 4 for y, and –1 for x.

GUIDED PRACTICE for Example 1

Write an equation in point-slope form of the line that passes through the point (– 1, 4) and has a slope of – 2 .

1.

Warm-Up ExercisesGraph an equation in point-slope form

EXAMPLE 2

y + 2 = (x – 3).2 3

Graph the equation.Hint

y-y1 = m(x-x1)


EXAMPLE 2

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

Plot the point (3, – 2). Find a secondpoint on the line using the slope.Draw a line through both points.


EXAMPLE 2

y – 1 = (x – 2)–Graph the equation.2.


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.

Warm-Up ExercisesUse point-slope form to write an equation

EXAMPLE 3

Write an equation in point-slope form of the line shown.


EXAMPLE 3

SOLUTION

STEP 1

= y1 – y2x1 – x2

m = 3 –1– 1 –1 =

2– 2

= – 1

Find the slope of the line.


EXAMPLE 3

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)

STEP 2Write the equation in point-slope form. You can us either given point.

CHECKCheck that the equations are equivalent by writing them in slope-intercept form.y – 3 = –x – 1

y = –x + 2y – 1 = –x + 1

y = –x + 2


EXAMPLE 3

STEP 1

= y2 – y1x2 – x4

m = 4 –3 4 –2 =

1 2

Find the slope of the line.


Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4).

3.


EXAMPLE 3

Method 1 Method 2

Use (2, 3) Use (4, 4)y – y1 = m(x – x1) y – y1 = m(x – x1)

STEP 2Write the equation in point-slope form. You can us either given point.


y – 3 = (x – 2)1

2y – 4 = (x – 4)

1 2

Warm-Up ExercisesSolve a multi-step problemEXAMPLE 4

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.

Warm-Up ExercisesSolve a multi-step problem

EXAMPLE 4

SOLUTION

STEP 1

Identify the rate of change and a data pair. Let C be 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): (1 thousand stickers, $225)


EXAMPLE 4

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.

STEP 2

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.


EXAMPLE 4

Find the cost of 9000 stickers.

C = 80(9) + 145 = 865Substitute 9 for s. Simplify.

ANSWER

The cost of 9000 stickers is $865.

STEP 3

Warm-Up ExercisesWrite a real-world linear model from a table

EXAMPLE 5

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.

4 6 8

250

10Number of people

Cost (dollars) 550450350 650

12


EXAMPLE 5

SOLUTION

Find the rate of change for consecutive data pairs in the table.

STEP 1

650 – 55012 – 10 = 50

350 – 2506 – 4 = 50, 450 – 350

8 – 6 = 50, 550 – 45010 – 8 = 50,

Because the cost increases at a constant rate of $50 per person, the situation can be modeled by a linear equation.


EXAMPLE 5

STEP 2

Use point-slope form to write the equation. Let C be the cost (in dollars) and p be 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.

Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5

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.

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.

b. Which Company would charge you less for 9000 stickers?


SOLUTIONSTEP 1

Identify the rate of change and a data pair. Let C be the cost (in dollars) and s be the number of stickers (in thousands).

Rate of change, m: $60 per 1 thousand stickers

Data pair (s1, C1): (1 thousand stickers, $250)


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.

STEP 2

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.


Find the cost of 9000 stickers.

C = 60(9) + 190 = 730Substitute 9 for s. Simplify.

ANSWER

The cost of 9000 stickers is $730. The second Company would charge you less for9000 stickers.

STEP 3


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)

Cost (dollars)

1 4 5

0.37

10

2.441.291.06

12

2.90


SOLUTION

Find the rate of change for consecutive data pairs in the table.

STEP 1

2.90 – 2.4412 – 10

= 0.23

1.06 – 0.374 – 1 = 0.23, 1.29 – 1.06

5 – 4 = 0.23, 2.44 – 1.2910 – 5 = 0.23,

Because the cost increases at a constant rate of $0.23 per mail, the situation can be modeled by a linear equation.


STEP 2

Use point-slope form to write the equation. Let C be the cost (in dollars) and w be 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 = 0.23w +0.14 Solve for C.

C – 0.37 = 0.23w – 0.23 Simplify

Warm-Up

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