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 in table 2 g(x)
Table 1: x:0, 1, 2 f(x):-3, 6, -9 Table 2: x: 0, 1, 2 g(x):

Answer 1

To create the function \( g(x) \) from the parent function \( f(x) = -12x^2 + 21x - 3 \) by horizontally stretching it with a factor of \( k = \frac{1}{2} \), we need to apply a transformation. The general transformation for horizontally stretching a function by a factor of \( k \) involves substituting \( x \) with \( \frac{x}{k} \).

In this case, since \( k = \frac{1}{2} \), the transformation will be:

\[
g(x) = f\left(\frac{x}{\frac{1}{2}}\right) = f(2x)
\]

Now, we need to compute \( g(x) \) for the same values of \( x \) as in Table 1, which are \( 0, 1, \) and \( 2 \).

### Step 1: Calculate \( g(0) \)
\[
g(0) = f(2 \cdot 0) = f(0) = -12(0)^2 + 21(0) - 3 = -3
\]

### Step 2: Calculate \( g(1) \)
\[
g(1) = f(2 \cdot 1) = f(2)
\]
Compute \( f(2) \):
\[
f(2) = -12(2)^2 + 21(2) - 3 = -12(4) + 42 - 3 = -48 + 42 - 3 = -9
\]
So:
\[
g(1) = -9
\]

### Step 3: Calculate \( g(2) \)
\[
g(2) = f(2 \cdot 2) = f(4)
\]
Compute \( f(4) \):
\[
f(4) = -12(4)^2 + 21(4) - 3 = -12(16) + 84 - 3 = -192 + 84 - 3 = -111
\]
So:
\[
g(2) = -111
\]

### Summary of Values
Now we can complete Table 2:

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

Thus, the completed Table 2 is:
- \( g(0) = -3 \)
- \( g(1) = -9 \)
- \( g(2) = -111 \)