Given g(x) = 2x2 +1 , find g(3)

g(3)=2(3)2 +1

g(3)=2(9)+1

g(3)=19

Given f(x) = 3x - 1, find 4[f(4) – f(2)]

Given h(x) = 5+5x, find [f(4) – f(1)](-2)

3.3 Linear FunctionsThe student will be able to:

1. identify linear functions.

2. write equations in standard form.

3. Graph linear functions.

Indicators: Designed by Skip Tyler, Varina High School

Vocabularies

• A linear equation is the equation of a line. • An x-intercept is a point on the graph where y is

zero. In other words, an x-intercept is a point in the equation where the y-value is zero.

• A y-intercept is a point on the graph where x is zero. In other words, a y-intercept is a point in the equation where the x-value is zero.

• Values of x for which f(x)=0 are called zeros. For a linear function, the zero is its x-intercept.

Standard FormAx + By = C

A, B, and C have to be integers

An equation is LINEAR (the graph is a straight line) if it can be written in standard form.

This form is useful for graphing (later on…).

Determine whether each equation is a linear equation.

1) 4x = 7 + 2y

Can you write this in the form

Ax + By = C?

4x - 2y = 7

A = 4, B = -2, C = 7

This is linear!

2) 2x2 - y = 7Can you write it in standard form?

NO - it has an exponent!Not linear

3) x = 12x + 0y = 12

A = 1, B = 0, C = 12Linear

Determine whether each equation is a linear equation.

Here’s the cheat sheet! An equation that is linear does NOT contain the following:

1. Variables in the denominator

2. Variables with exponents

3. Variables multiplied with other variables.

xy = 12

32y

x

2 3y x

Is this equation linear?

1. Yes

2. No

4 3x y

Standard Formx – 4y = 3

Is this equation linear?

1. Yes

2. No

29 4y x

Exponents are not allowed!

Is this equation linear?y = -3

1. Yes

2. No

Standard Form0x + y = -3

Review of the definition.

• An x-intercept is a point on the graph where y is zero. In other words, an x-intercept is a point in the equation where the y-value is zero.

• A y-intercept is a point on the graph where x is zero. In other words, a y-intercept is a point in the equation where the x-value is zero.

• Values of x for which f(x)=0 are called zeros. For a linear function, the zero is its x-intercept.

Determine the x-intercept, y-intercept, and zero of the linear equation y=10x-1.

x-intercept:•Set y=0•Solve for x

0=10x-1 +1 +1 1=10xx=1/10

y-intercept:•Set x=0•Solve for y

y=10(0)-1

y=-1

Determine the x-intercept, y-intercept, and zero of the linear equation y=10x-1.

x-intercept: y-intercept:

Find the x-intercept, y-intercept, and zero for the graph of the line.

Solution:The x-intercept of this graph is 4. The coordinates of the point at the x-intercept is (4, 0).The y-intercept of this graph is 2. The coordinates of the point at the y-intercept is (0, 2).

Find the x-intercept, y-intercept, and zero for the graph of the line.

x-intercept: -3y-intercept: 3zero: 3

Work on the following problems:

• P 158 #3, 12-19 all,

Graph an equation by making a table

Graphing Steps

1) Isolate the variable (solve for y).

2) Make a t-table. If the domain is not given, pick your own values.

3) Plot the points on a graph.

4) Connect the points.

1) Review: Solve for y 2x + y = 4

1. Draw “the river”

2. Subtract 2x from both sides

- 2x - 2x

y = -2x + 42) Solve for y: 4x + 2y = -6

1. Subtract 4x

2. Simplify

3. Divide both sides by 2

4. Simplify

- 4x - 4x

2y = -4x - 6

2 2

y = -2x - 3

3) Solve for y: x - 3y = 6

1. Subtract x

2. Simplify

3. Divide both sides by -3

4. Simplify

- x - x

-3y = -x + 6

-3 -36

3

xy

2

3

xy

or

4) Review: Make a t-tableIf f(x) = 2x + 4, complete a table using the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)

2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)

2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)-2

-1

0

1

2

ordered pair

5) Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)

-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-1

0

1

2

Bonus questions!What is the x-intercept?

(2, 0)What is the y-intercept?

(0, 6)Does the line increase or decrease?

Decrease

5) Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6

3. Plot the points(-2,12), (-1,9), (0,6), (1,3), (2,0)

4. Connect the points.

5) Given the domain {-2, -1, 0, 1, 2}, graph 4x + 2y = 6

1. .

2. .

3. .

4. .

Which is the graph of y = x – 4?

• P 158 36-37, 39-43 odds ( might change due to time consuming)