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
. Find g(x) values

Table 1: x-values: 0, 1, 2 and f(x) values=-3, 6, -9
Table 2: x-values: 0, 1, 2 and g(x) values:

Answer 1

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

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

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

Now we will compute \( g(x) \) for the x-values provided in Table 2: 0, 1, and 2.

### Calculate

1. **For \( x = 0 \):**
\[
g(0) = f(2 \times 0) = f(0) = -3
\]

2. **For \( x = 1 \):**
\[
g(1) = f(2 \times 1) = f(2)
\]
We need to compute \( f(2) \):
\[
f(2) = -12(2^2) + 21(2) - 3 = -12(4) + 42 - 3 = -48 + 42 - 3 = -9
\]
Thus, \( g(1) = -9 \).

3. **For \( x = 2 \):**
\[
g(2) = f(2 \times 2) = f(4)
\]
We compute \( f(4) \):
\[
f(4) = -12(4^2) + 21(4) - 3 = -12(16) + 84 - 3 = -192 + 84 - 3 = -111
\]
Thus, \( g(2) = -111 \).

### Summary of Results

The values for \( g(x) \) are as follows:

| Table 2: x-values | g(x) values |
|--------------------|--------------|
| 0 | -3 |
| 1 | -9 |
| 2 | -111 |

So the completed table would look like this:

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