Question

1. using $f(x)=x^{2}$ as a guide, describe the transformation of each function.

$g(x)=4(x + 3)^{2}-2$

1. the parent function $f(x)=x^{2}$ is vertically stretched by a factor of 4 and then translated 2

units left and 4 units up to create $g(x)$. then write $g(x)$
$g(x)=$

1. the average gas mileage $m$ in mpg for an average sedan is modeled by

$m_{sed}(s)=-0.01(s - 30)^{2}+28$, where $s$ is the cars speed in mph. the average gas
mileage for an suv is modeled by. $m_{suv}(s)=-0.01(s - 30)^{2}+12$
what kind of transformation describes this change, and what does this transformation
mean?
transformation:
meaning:

Explanation:

13.

For a function \(y = a(x - h)^{2}+k\) compared to \(y=x^{2}\):

• The factor \(a = 4\) means a vertical stretch by a factor of \(4\).
• The value \(h=- 3\) (since the function is \((x + 3)\)) means a horizontal translation \(3\) units to the left.
• The value \(k=-2\) means a vertical translation \(2\) units down.

For the functions \(m_{\text{sed}}(s)=-0.01(s - 30)^{2}+28\) and \(m_{\text{SUV}}(s)=-0.01(s - 30)^{2}+12\), comparing them to the form \(y=a(x - h)^{2}+k\).

• The transformation is a vertical translation. Here, \(a=-0.01\) and \(h = 30\) are the same for both functions, and \(k\) changes from \(28\) to \(12\).
• The vertical translation formula is \(y=f(x)+k\). If \(k_{1}=28\) and \(k_{2}=12\), then \(y_{2}=y_{1}+(12 - 28)\), so it is a vertical translation \(16\) units down.
• In the context of gas - mileage, since the functions model gas mileage \(m\) (in mpg) as a function of speed \(s\) (in mph), a lower \(k\) value (vertical translation down) means that for the same speed \(s\), the SUV has a lower gas mileage than the sedan.

Answer:

The function \(g(x)=4(x + 3)^{2}-2\) is a vertical stretch by a factor of \(4\), a horizontal translation \(3\) units to the left, and a vertical translation \(2\) units down of the parent function \(f(x)=x^{2}\).

14.

Step 1: Vertical Stretch

For the parent function \(f(x)=x^{2}\), a vertical stretch by a factor of \(a = 4\) gives \(y = 4x^{2}\).

Step 2: Horizontal Translation

A horizontal translation \(h\) units. Since it is \(2\) units to the left (\(h=-2\)), the function becomes \(y = 4(x+2)^{2}\).

Step 3: Vertical Translation

A vertical translation \(k\) units. Since it is \(4\) units up (\(k = 4\)), the function \(g(x)=4(x + 2)^{2}+4\).