© A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 3.

© A Very Good Teacher 2007

Exit Level

TAKS Preparation UnitObjective 3


Interpreting Linear Functions Functions can be represented in different ways:

y = 2x + 3 means the same thing as f(x) = 2x + 3

Linear Functions must have a slope (rate of change) and a y intercept (initial value).

In a function… the slope is the constant (number) next to the

variable the y intercept is the constant (number) by itself

3, Ac1A


Interpreting Linear Functions, cont…

• Example: Identify the situation that best represents the amount f(n) = 425 + 50n.

Slope (rate of change) =

Y intercept (initial value) =

50425

Find an answer that has:

425 as a non-changing value and

50 as a recurring charge every month, every year, etc…

Something like Joe has $425 in his savings account and he adds $50 every month.

3, Ac1A


Converting Tables to Equations• When given a table of values, USE STAT!• Example: What equation describes the relationship

between the total cost, c, and the number of books, b?

b c

10 75

15 100

20 125

25 150Answer: c = 5x + 25

3, Ac1C


Converting Graphs to Equations• Make a table of values

• Then, use STAT!

• Example: Which linear function describes the graph shown below?

3, Ac1C

x y

Answer: y = -.5x + 4

-2 5

0 4

2 3

4 2


Converting Equation to Graph

• Graph the function in y =

• Example: Which graph best describes the function y = -3.25x + 4?

Find an answer that has the same y intercept and x intercept as the calculator graph.

3, Ac1C


Equations that are in Standard Form• Sometimes your equations won’t be in

y = mx + b form.

• They will be in standard form: Ax + By = C• You must convert them to use the calculator!

Example: 3x + 2y = 12Step 1: Move the x -3x -3x

2y = -3x + 12 Step 2: Divide everything by the number in front of y

2 2 2

36

2y x

3, Ac1C


Slope and Rate of Change (m)• Slope and rate of change are the same thing!

• They both indicate the steepness of a line.

• Three ways to find the slope of a line:

By Formula: By Counting: By Looking:

2 1

2 1

y ym

x x

rise

mrun

y x bm

You must have 2 points

on a line

You must have a graph

You must have an equation

3, Ac2A


Slope and Rate of Change (m), cont…• By Formula:

• Find two points on the graph (they won’t be given to you)

3, Ac2A

2 1

2 1

y ym

x x

(0, 4) and (2, 3)1x 2x1y 2y

1

2

3 4

2 0


Slope and Rate of Change (m), cont…

• By Counting

• Find two points on the graph

3, Ac2A

risem

run

Down 2

Right 4

2

4

1

2


Slope and Rate of Change (m), cont…• By Looking

• The equation won’t be in y = mx + b form

• You’ll have to change it• If in Standard Form use Process on Slide 7• If in some other form, you’ll have to work it out…

3, Ac2A

Example: What is the rate of change of the function

4y = -2(x – 24)? Try to get rid of any parentheses and get the y by itself (isolated).

4y = -2x + 244 4 4

16

2y x

1

2m


Slope and Rate of Change (m), cont…

• Special Cases

• Horizontal lines line y = 4

3, Ac2A

• Vertical lines like x = 4

Have slope of zero, m = 0

Have slope that is undefined


m and b in a Linear Function• Changes to m, the

slope, of a line effect its steepness

3, Ac2C

• Changes to b, the y intercept, of a line effect its vertical position (up or down)

y = 1x + 0

y = 3x + 0

y = 1/3 x + 0

y = 1x + 0

y = 1x + 3

y = 1x - 4


m and b in a Linear Function, cont…

• Parallel Lines have equal slope (m)

y = ¼ x – 3 and y = ¼ x + 6

• Perpendicular Lines have opposite reciprocal slope (m)

y = ¼ x – 5 and y = -4x + 15

• Lines with the same y intercept will have the same number for b

y = ¾ x – 9 and y = 5x – 9

3, Ac2C


Linear Equations from Points• Make a table

• USE STAT

• Example: Which equation represents the line that passes through the points (3, -1) and (-3, -3)?

x y

3 -1

-3 -3

Answer: 1

23

y x

3, Ac2D


Intercepts of Lines• To find the

intercepts from a graph… just look!

• The x intercept is where a line crosses the x axis

• The y intercept is where a line crosses the y axis

3, Ac2E

(4, 0)

(0, 2)


Intercepts of Lines, cont…• To find intercepts from equations, use your

calculator to graph them

• Example: Find the x and y intercepts of 4x – 3y = 12.

-4x -4x

-3y = -4x + 12-3 -3 -3

44

3y x

x intercept: (3, 0)

y intercept: (0, -4)

3, Ac2E


Direct Variation• Set up a proportion!

• Make sure that similar numbers appear in the same location in the proportion

• Example: If y varies directly with x and y is 16 when x is 5 what is the value of x when y = 8?

3, Ac2F

16x = 5(8)

16x = 4016 16

x = 2.5

y y

x x 16 8

5 x


Direct Variation, cont…• To find the constant of variation use a

linear function (y = kx) and find the slope

• The slope, m, is the same thing as k• Example: If y varies directly with x and y = 6

when x = 2, what is the constant of variation?

y = kx

6 = k(2)2 2

3 = k

The equation for this situation would be y = 3x

3, Ac2F

© A Very Good Teacher 2007 Exit Level TAKS Preparation Unit Objective 3.

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