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High School Math : Polynomial Functions

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Example Questions

Example Question #1 : Understanding Polynomial Functions

It took Jack 25 minutes to travel 14 miles, what was Jack's average speed in mph?

Possible Answers:

$2.4\ mph$

$1.78\ mph$

$33.6\ mph$

$30.21\ mph$

Correct answer:

$33.6\ mph$

Explanation:

$speed=\frac{distance}{time}$

$speed=\frac{14\ miles}{25\ minutes }$

* We have to change the time from minutes to hours, there are 60 minutes in one hour.

$speed= \frac{14\ miles}{25\ minutes} * \frac{60\ minutes}{1\ hour}$

$speed= 33.6\ mph$

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Example Question #2 : Understanding Polynomial Functions

Let $f(x)=3x^{2}-2$ and g(x)= x+2 . Evaluate f(g(3)) .

Possible Answers:

108

Correct answer:

Explanation:

Substitute into g(x) , and then substitute the answer into f(x) .

g(x)= x+2

g(3)=3+2=5

$f(x)= 3x^{2}-2$

$f(5) =3(5)^{2}-2= 75 -2 = 73$

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Example Question #2 : Understanding Polynomial Functions

Solve the following system of equations:

-x-3y= 4

2x+5y=7

Possible Answers:

$\left ( 1,-5 \right )$

Infinite solutions.

$\left ( 2,4 \right )$

$\left ( 41,-15 \right )$

$\left ( -4,10 \right )$

Correct answer:

$\left ( 41,-15 \right )$

Explanation:

We will solve this system of equations by Elimination. Multiply both sides of the first equation by 2, to get:

-2x-6y=8

Then add this new equation, to the second original equation, to get:

-y = 15

or

y=-15

Plugging this value of back into the first original equation, gives:

-x-3(-15) = 4

-x = 4- 45

or

x = 41

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Example Question #1 : Transformations Of 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 4

horizontal compression 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

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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