Transformations of Linear Functions

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Algebra II › Transformations of Linear Functions

Questions 1 - 10

1

Shift the graph down four units. What is the new equation?

$y=\frac{3}{2}x-2$

$y=\frac{3}{2}x$

$y=\frac{3}{2}x+1$

$y=\frac{3}{2}x-1$

$y=\frac{3}{2}x-5$

Explanation

Rewrite this equation in slope intercept form .

Add on both sides.

The equation becomes:

Divide by two on both sides.

$\frac{2y }{2}= \frac{3x+4}{2}$

The equation in slope intercept form is: $y=\frac{3}{2}x+2$

Shifting this equation down four units means that the y-intercept will be decreased four units.

The answer is: $y=\frac{3}{2}x-2$

2

The line is shifted right 5 units. What must be the new equation?

$\textup{The answer is not given.}$

Explanation

If the line is shifted right 5 units, we will need to replace the x-variable of the equation with the quantity .

Simplify by distribution.

The answer is:

3

Shift the line up six units. What is the new equation?

Explanation

Add six to the equation since a vertical shift will increase the y-intercept by six units.

Simplify this equation by distribution.

The answer is:

4

Translate the function up two units. What is the y-intercept of the new equation?

$2\frac{1}{2}$

$2\frac{1}{4}$

$1\frac{1}{2}$

$1\frac{1}{4}$

$\frac{1}{2}$

Explanation

The equation given is currently in standard form.

Rewrite the equation in slope-intercept form, .

Subtract on both sides of .

Divide by two on both sides.

$\frac{2y}{2}=\frac{-2x+1}{2}$

Simplify the fractions and split the right fraction into two parts.

The equation in slope-intercept form is: $y=-x+\frac{1}{2}$

Apply the translation. If this line is shifted up two units, the y-intercept will be added two.

$\frac{1}{2}+2 = 2\frac{1}{2}$

The answer is: $2\frac{1}{2}$

5

Shift the line to the left 8 units. What is the new equation?

Explanation

To shift the line left 8 units, we will need to replace the x-value with .

Distribute the negative four through the binomial.

Combine like-terms.

The answer is:

6

Translate the equation down six units. What is the new equation?

Explanation

Use distribution to simplify the equation.

The equation in slope-intercept form is:

Shift the equation down six units means subtract the y-intercept by six.

The answer is:

7

Shift the graph $y=9-\frac{2}{9}x$ three units to the left. What's the new equation?

$y=-\frac{2}{9}x+\frac{25}{3}$

$y=-\frac{2}{9}x+3$

$y=-\frac{2}{9}x+\frac{1}{6}$

$y=-\frac{2}{9}x+\frac{31}{3}$

$y=-\frac{2}{9}x+\frac{29}{3}$

Explanation

In order to shift an equation to the left three units, the x-variable will need to be replaced with the quantity of . This shifts all points left three units.

$y=9-\frac{2}{9}(x+3)$

Simplify the equation.

$y=9-\frac{2}{9}x-\frac{2}{3} = \frac{27}{3}-\frac{2}{9}x-\frac{2}{3}$

The answer is: $y=-\frac{2}{9}x+\frac{25}{3}$

8

Translate the following function left three units: Write the new equation.

Explanation

When the graph is shifted three units to the left, the x-variable of the equation will need to be replaced with .

Replace the x term with the new term.

Simplify this equation.

Combine like-terms.

The answer is:

9

Shift the equation $y=-\frac{1}{3}x-1$ to the left two units. What is the new equation?

$y=-\frac{1}{3}x-\frac{5}{3}$

$y=-\frac{1}{3}x-\frac{1}{3}$

$y=-\frac{1}{3}x+\frac{1}{3}$

$y=-\frac{1}{3}x+\frac{2}{3}$

$y=-\frac{1}{3}x-\frac{2}{3}$

Explanation

If the linear function is shifted left two units, the x-variable must be replaced with the quantity of .

$y=-\frac{1}{3}(x+2)-1$

Simplify the equation by distribution.

$y=-\frac{1}{3}x-\frac{2}{3}-1= -\frac{1}{3}x-\frac{2}{3}-\frac{3}{3}$

Combine like terms.

The answer is: $y=-\frac{1}{3}x-\frac{5}{3}$

10

Translate the equation left four units. What is the new equation?

Explanation

To shift the line left four units, we will need to replace the x-variable with the quantity of:

Replace this term in the original equation.

Use distribution to simplify.

The answer is:

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