Polynomial Functions - Math

Since the equation given in the question is based off of the parent function , we can write the general form for transformations like this:

determines the vertical stretch or compression factor.

• If is greater than 1, the function has been vertically stretched (expanded) by a factor of .
• If is between 0 and 1, the function has been vertically compressed by a factor of .

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 is greater than 1, the function has been horizontally compressed by a factor of .
• If is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of .

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.

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

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

or

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

or

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

Substitute into , and then substitute the answer into .

First, we need to get this function into a more standard form.

Now we can see that while the function is being horizontally compressed by a factor of 2, it's being translated 3 units to the right, not 6. (It's also being vertically stretched by a factor of 4, of course.)

In this problem, the in the equation becomes --> .

This simplifies to , or

.

By definition, , so

By definition, , so

To transform the function horizontally, we must make an addition or subtraction to the input, x. Because we are asked to move the function to the left, we must add the number of units we are moving. This is the opposite of what one would expect, but if we are inputting values that are to the left of the original, they are less than what would have originally been. So, to counterbalance this, we add the units of the transformation.

For our function being transformed five units to the left, we get

To transform a function we use the following formula,

where h represents the horizontal shift and v represents the vertical shift.

In this particular case we want to shift to the left five units,

and vertically up two units,

.

Therefore, the transformed function becomes,

.

To transform a function horizontally, we must add or subtract the units we transform to x directly. To move left, we add units to x, which is opposite what one thinks should happen, but keep in mind that to move left is to be more negative. To flip a function, the entire function changes in sign.

After making both of these changes, we get

Expand the binomial.

Multiply negative by this quantity.

The polynomial in standard form is:

Shifting this graph up one will change the y-intercept by adding one unit.

The answer is:

Shifting this parabola up two units requires expanding the binomial.

Use the FOIL method to simplify this equation.

Shifting this graph up two units will add two to the y-intercept.

The answer is:

We will need to determine the equation of the parabola in standard form, which is:

Use the FOIL method to expand the binomials.

Shifting this up two units will add two to the value of .

The answer is:

Since the equation given in the question is based off of the parent function , we can write the general form for transformations like this:

determines the vertical stretch or compression factor.

• If is greater than 1, the function has been vertically stretched (expanded) by a factor of .
• If is between 0 and 1, the function has been vertically compressed by a factor of .

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 is greater than 1, the function has been horizontally compressed by a factor of .
• If is between 0 and 1, the function has been horizontally stretched (expanded) by a factor of .

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.

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

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

or

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

or

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

Substitute into , and then substitute the answer into .

First, we need to get this function into a more standard form.

Now we can see that while the function is being horizontally compressed by a factor of 2, it's being translated 3 units to the right, not 6. (It's also being vertically stretched by a factor of 4, of course.)

In this problem, the in the equation becomes --> .

This simplifies to , or

.