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High School Math : Transformations of Polynomial Functions

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Example Questions

Example Question #1 : Polynomial Functions

List the transformations that have been enacted upon the following equation:

$f(x)=4[6(x-3)]^{4}-7$

Possible Answers:

vertical stretch by a factor of 4

horizontal stretch by a factor of 6

vertical translation 7 units down

horizontal translation 3 units right

vertical stretch by a factor of 1/4

horizontal compression by a factor of 1/6

vertical translation 7 units down

horizontal translation 3 units right

vertical compression by a factor of 4

horizontal stretch by a factor of 6

vertical translation 7 units down

horizontal translation 3 units right

vertical stretch by a factor of 4

horizontal compression by a factor of 6

vertical translation 7 units down

horizontal translation 3 units left

vertical stretch by a factor of 4

horizontal compression by a factor of 6

vertical translation 7 units down

horizontal translation 3 units right

Correct answer:

vertical stretch by a factor of 4

horizontal compression by a factor of 6

vertical translation 7 units down

horizontal translation 3 units right

Explanation:

Since the equation given in the question is based off of the parent function $y = x^{4}$ , we can write the general form for transformations like this:

$g(x) = a[b(x-c)^{4}]+d$

determines the vertical stretch or compression factor.

If $\left | a \right |$ is greater than 1, the function has been vertically stretched (expanded) by a factor of .
If $\left | a \right |$ is between 0 and 1, the function has been vertically compressed by a factor of $\frac{1}{\left | a \right |}$ .

In this case, is 4, so the function has been vertically stretched by a factor of 4.

determines the horizontal stretch or compression factor.

If $\left | b \right |$ is greater than 1, the function has been horizontally compressed by a factor of .
If $\left | b \right |$ is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of $\frac{1}{\left | b \right | }$ .

In this case, is 6, so the function has been horizontally compressed by a factor of 6. (Remember that horizontal stretch and compression are opposite of vertical stretch and compression!)

determines the horizontal translation.

If is positive, the function was translated units right.
If is negative, the function was translated units left.

In this case, is 3, so the function was translated 3 units right.

determines the vertical translation.

If is positive, the function was translated units up.
If is negative, the function was translated units down.

In this case, is -7, so the function was translated 7 units down.

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