Question

the function g(x) is created by horizontally stretching the parent function, f(x)=-12x^2 + 21x -3 with a k factor of 1/2. complete the table to show how the function has changed.
x f(x)
0, -3
1, 6
2, -9
table 2
x g(c)
0, ?
1, ?
2, ?

Answer 1

To create the function \( g(x) \) from the parent function \( f(x) = -12x^2 + 21x - 3 \) by horizontally stretching it with a \( k \) factor of \( \frac{1}{2} \), we substitute \( x \) with \( 2x \) in the function \( f(x) \). This gives us:

\[
g(x) = f(2x) = -12(2x)^2 + 21(2x) - 3
\]

Now, let's simplify \( g(x) \):

\[
g(x) = -12(4x^2) + 42x - 3
\]

\[
g(x) = -48x^2 + 42x - 3
\]

Next, we will find the values of \( g(0) \), \( g(1) \), and \( g(2) \).

1. **For \( x = 0 \)**:
\[
g(0) = -48(0)^2 + 42(0) - 3 = -3
\]

2. **For \( x = 1 \)**:
\[
g(1) = -48(1)^2 + 42(1) - 3 = -48 + 42 - 3 = -9
\]

3. **For \( x = 2 \)**:
\[
g(2) = -48(2)^2 + 42(2) - 3 = -48(4) + 84 - 3 = -192 + 84 - 3 = -111
\]

Now we can create the second table \( g(x) \):

\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
0 & -3 \\
1 & -9 \\
2 & -111 \\
\hline
\end{array}
\]

So, the completed table will look like this:

- \( x = 0 \): \( g(0) = -3 \)
- \( x = 1 \): \( g(1) = -9 \)
- \( x = 2 \): \( g(2) = -111 \)

Thus, the second table is:

\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
0 & -3 \\
1 & -9 \\
2 & -111 \\
\hline
\end{array}
\]