Question

F(x)=-∣x∣2-3

Answer 1

Answer:

$f^\prime(x)=-2|x-1|-3$

Step-by-step explanation:

Write the equation for f(x)=IxI but vertically stretch it by a factor of 2, flip it over the x axis, shift it down 3 units, shift it 1 unit to the right

We have the (parent absolute value) function:

$f(x)=|x|$

And we want to write a new function which represents the previous function: 1) Vertically stretched by a factor of 2, 2) flipped over the x-axis, 3) shifted down 3 units, and 4) shifted 1 unit to the right.

1)

To vertically stretch/compress a function, we multiply the function by a constant a.

• If a is greater than 1, it is a vertical stretch.

• If a is less than 1 (and greater than 0), it is a vertical compression.

We want to vertically stretch our function by 2. So, we will multiply our function by 2. Therefore:

$f^\prime(x)=2|x|$

2)

We want to flip the function over the x-axis.

To flip a function over either axis:

• Multiply by negative 1 to flip the function over the x-axis.
• Substitute x for -x to flip the function over the y-axis.

Since we want to flip our function over the x-axis, we will multiply it by -1. So:

$f^\prime(x)=-2|x|$

3)

We want to shift our function down by 3 units.

To shift a function vertically, we simply add the vertical shift to our function.

Since we are shifting downwards by 3 units, we will add -3 to our function. So:

$f^\prime(x)=-2|x|-3$

4)

Finally, we want to shift our function 1 unit to the right.

To shift a function horizontally, we replace the x with (x-n), where n is the horizontal shift.

Since we are shifting our function to the right, n=1. Therefore, we will replace x with (x-1). So, our function becomes:

$f^\prime(x)=-2|x-1|-3$

And we have our final function.

Note: In general, try doing the vertical/horizontal stretches first, x/y-axis reflections second, and then vertical/horizontal shifts last to avoid mistakes.