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

x g(x)
0 __
1 ___
2 ___

Answer 1

To find the values of \( g(x) \) after applying a horizontal stretch to the function \( f(x) \), we use the fact that a horizontal stretch by a factor of \( k \) affects the input \( x \) in the function \( f \). Specifically, the transformation is given by:

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

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

\[
g(x) = f(2x)
\]

Now we will calculate \( g(x) \) for \( x = 0 \), \( x = 1 \), and \( x = 2 \):

1. **For \( x = 0 \)**:
\[
g(0) = f(2 \cdot 0) = f(0)
\]
Since \( f(0) = -3 \), we have:
\[
g(0) = -3
\]

2. **For \( x = 1 \)**:
\[
g(1) = f(2 \cdot 1) = f(2)
\]
From the original table, \( f(2) = -9 \), so:
\[
g(1) = -9
\]

3. **For \( x = 2 \)**:
\[
g(2) = f(2 \cdot 2) = f(4)
\]
To find \( f(4) \), we will evaluate the function:
\[
f(4) = -12(4^2) + 21(4) - 3
\]
\[
= -12(16) + 84 - 3
\]
\[
= -192 + 84 - 3
\]
\[
= -192 + 81
\]
\[
= -111
\]
Thus:
\[
g(2) = -111
\]

So, summarizing the results, we have the following completed table:

| x | f(x) | g(x) |
|---|------|-------------|
| 0 | -3 | -3 |
| 1 | 6 | -9 |
| 2 | -9 | -111 |

Therefore, the values of \( g(x) \) are:

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