Comparing Multiple Representations9-4

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

Comparing Multiple Representations9-4

Warm Up1. Find the slope of a line through points (3, 4)

and (6, 2).

2. The slope of a line is 2 and the y-intercept is

10. What is the slope-intercept form of the

line?

y = 2x + 10

-2 3

Comparing Multiple Representations9-4

Problem of the Day

Chicken salad costs $5 per person and egg salad costs $4 per person. Write these two options as linear functions and compare them in terms of the context.

Chicken sal f(x) = 5x and egg sal g(x) = 4x; f has a greater rate of change (cost per person).

Comparing Multiple Representations9-4

Learn to compare linear functions represented in different ways.

Comparing Multiple Representations9-4

Remember that slope is rise (change in y) divided by run (change in x) or ‘rise over run’.

Helpful Hint

Comparing Multiple Representations9-4

Find and compare the slopes for the linear functions f and g.

f(x) = 10x – 5

Additional Example 1: Comparing Slopes

Find the slope of f.

Function f is written in slope-intercept form.

f(x) = mx + b

f(x) = 10x – 5

The slope of f is 10.

x –2 –1 0 1

g(x) 0 5 10 15

Comparing Multiple Representations9-4

Additional Example 1: Continued

Find the slope of g.

The slope of g is 5.

The slope f is greater than the slope of g.

m =y2 – y1x2 – x1

10 – 50 – (–1)

= = 5

Comparing Multiple Representations9-4

Find and compare the slopes for the linear functions f and g.

f(x) = 8x + 4

Check It Out: Example 1

x –2 –1 0 1

g(x) 0 8 16 24

Find the slope of f.

Function f is written in slope-intercept form.

f(x) = mx + b

f(x) = 8x + 4

The slope of f is 8.

Comparing Multiple Representations9-4

Continued: Check It Out Example 1

Find the slope of g.

The slope of g is 8.

The slope f is the same as the slope of g.

m =y2 – y1x2 – x1

16 – 80 – (–1)

= = 8

Comparing Multiple Representations9-4

Additional Example 2: Comparing Intercepts

Find the y-intercept of f.

When x = 0, f(x) = –6.

The y-intercept of f is –6.

Find the y-intercept of g.

Find and compare the y-intercepts for the linear functions f and g.

The graph of g crosses the y-axis at about –5.

The y-intercept of g is greater than the y-intercept of f.

x 0 1 2 3

f(x) -6 -4 -2 0

Comparing Multiple Representations9-4

Check It Out: Example 2

Find the y-intercept of f.

When x = 0, f(x) = 3.

The y-intercept of f is 3.

Find the y-intercept of g.

The graph of g crosses the y-axis at about 3.5.

The y-intercept of f is greater than the y-intercept of g.

x –1 0 1 2

f(x) 0 3 6 9

Comparing Multiple Representations9-4

Rate of change is given by the slope and initial value is given by the y-intercept of a linear function.

Remember!

Comparing Multiple Representations9-4

Additional Example 3: Application Fred and Gene are hang gliding. Fred is 700 feet above the ground and descending at 15 ft/s. Gene is descending as shown in the table. Interpret the rates of change and initial values of the linear functions in terms of the situations they model.

f(x) = –15x + 700

Fred’s descent.

f(x) = mx + b

f(x) = –15x +700 The rate of change is –15.

The initial value is 700.

Time (sec) 0 1 2 3

Height (ft) 575 565 555 545

Comparing Multiple Representations9-4

Additional Example 3: Continued

The rate of change is –10.

Gene’s descent.

The initial value is 575.

Gene’s rate of descent (10 ft/sec) is 5 ft/s less than Fred’s. Fred is 125 feet higher than Gene when they start.

m =y2 – y1x2 – x1

= = –10565 – 5751 – 0

Comparing Multiple Representations9-4

Check It Out: Example 3Fred and Gene are hang gliding. Fred is 600 feet above the ground and descending at 10 ft/s. Gene is descending as shown in the table. Interpret the rates of change and initial values of the linear functions in terms of the situations they model.

f(x) = –10x + 600

Time (sec)

0 1 2 3

Height (ft)

480 465 450 435

Comparing Multiple Representations9-4

Continued: Check It Out Example 3

Fred’s descent.

f(x) = mx + b

f(x) = –10x +600

The rate of change is –10.

The initial value is 600.

Comparing Multiple Representations9-4

Continued: Check It Out Example 3

The rate of change is –15.

Gene’s descent.

The initial value is 480.

Fred’s rate of descent (10 ft/sec) is 5 ft/s less than Gene’s. Fred is 120 feet higher than Gene when they start.

m =y2 – y1x2 – x1

= = –15465 – 4801 – 0

Comparing Multiple Representations9-4

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Comparing Multiple Representations9-4

Use the function and graph forItems 1–2.f(x) = -2x + 3

1. Find and compare the slopes for the linear functions f and g.

2. Find and compare the y-intercepts for the linear functions f and g.

Lesson Quiz : Part I

f: 3, g: 4; y-int. g > y-int. f

f: -2, g: -1.5; slope g > slope f

Comparing Multiple Representations9-4

3. A whale comes to the surface of the water at 80 ft/min from a depth of 1000 ft. Another whale

ascends as shown in the table. Find and compare the rates of change and initial values of the linear functions in terms of the situations they model

f(x) = 80x - 1000

Lesson Quiz : Part II

Whale #2 started at -800 ft, which is 200 ft higher than Whale #1. Whale #2 ascends at 100 ft /min, which is 20 ft/min faster than Whale #1.

Time (min)

0 1 2 3

Depth (ft)

-800 -700 -600 -500

Comparing Multiple Representations9-4

1. What slope is greater?

f(x) = –3x + 5

A. f

B. g

Lesson Quiz for Student Response Systems

Comparing Multiple Representations9-4

2. Which has the greater y-intercept?

f(x) = –3x + 5

A. f

B. g

Lesson Quiz for Student Response Systems