# Algebra II : Transformations of Linear Functions

## Example Questions

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### Example Question #1 : Transformations Of Linear Functions

Write the equation from the augmented matrix.

Explanation:

Do the first row first and use x and y to represent your variable.

### Example Question #2 : Transformations Of Linear Functions

Solve for  in the equation.

Explanation:

Solve for x by isolating the variable.

### Example Question #3 : Transformations Of Linear Functions

What is the equation of the line that intersects the point  and ?

Explanation:

We are only given the points the line intersects. This can be used to find the slope of the line, knowing that slope is rise/run, or change in /change in  or by the formula,

.

By substituting, we get

for the slope.

To find the  intercept, we can use the equation , where  ---> .

Since both given points are on the line, either can be used to solve for :

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### Example Question #4 : Transformations Of Linear Functions

Which line is perpendicular to the line ?

Explanation:

Lines that are perpendicular have negative reciprocal slopes. Therefore, the line perpendicular to  must have a slope of . Knowing that the slope of  is , only  has a slope of .

### Example Question #5 : Transformations Of Linear Functions

Which line would never intersect a line with the slope ?

Explanation:

This question is very simple once you realize that a line that will never intersect another line must have the same slope (parallel lines will never intersect). Therefore you must look for the choice that has a slope of . Each answer can be converted to the form  or by knowing that in the equation , the slope of the line is simply . In the correct answer, , the slope would be , which simplfies to .

*Note* the y-intercept is irrelevant to finding the correct answer.

### Example Question #6 : Transformations Of Linear Functions

If the equation  was shifted left three units and up one unit, what is the new equation of the line?

Explanation:

If the equation shifts left three units, the  term will become .

The equation shifting up one unit will change the y-intercept of the equation.

Rewrite the equation and distribute to simplify.

The correct equation is:

### Example Question #7 : Transformations Of Linear Functions

Write the equation of a line that is parallel and two points lower than the line .

Explanation:

Straight-line equations may be written in the slope-intercept form: .

In this form,  equals the slope of the line and  corresponds to the y-intercept.

The given line has a slope of  and a y-intercept of positive . A line that is parallel to another has the same slope. Therefore, the slope of the new line will have to be .

In order to shift a line down, you must change the y-intercept. Since we are moving the line down by  the y-intercept should be  because .

If we plug those values into the slope-intercept equation, then we have the answer: .

### Example Question #8 : Transformations Of Linear Functions

Given the equation , which of the following lines are steeper?

None of these.

Explanation:

Considering that slope (m) is defined as rise over run, you can look that the fractional slopes and determine which are steeper or more flat. For example,  is equivalent to up one and over 8 while  is equivalent to up one and over 10. As you can see the slope of the second line "runs" horizontally more than does the first slope and is therefore flatter. Based on this fact one can conclude that the larger the the slope, the steeper the line. So select the largest slope and this is the steepest line. In our case it is  because  is steeper (larger) than  (flatter and a smaller number).

### Example Question #9 : Transformations Of Linear Functions

The equation  is shifted eight units downward.  Write the new equation.

Explanation:

Rewrite the equation in slope-intercept format, .

Subtract two on both sides.

If the equation shifts eight units down, this means that the y-intercept, , would also subtracted eight units.

### Example Question #10 : Transformations Of Linear Functions

Which of the following describes the transformation of the function  from its parent function ?

Stretched vertically by a factor of 2 and translated 3 units to the left

Stretched vertically by a factor of 2 and translated 3 units down

Stretched vertically by a factor of 2 and translated 3 units to the right

Stretched vertically by a factor of 2 and translated 3 units up

Stretched vertically by a factor of 2 and translated 3 units to the right

Explanation:

The only differences among the answer choices is the translation. The translation of a function is determined by , which represents a horizontal translation h units to the right and k units up. In this case, h = 3 and k = 0, which indicates a translation 3 units to the right.

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