Question

Use the tables to answer the question.

The function g(x)
is created by horizontally stretching the parent function, f(x)=−12x2+21x−3
, with a k factor of 1/2
. Complete the table to show how the function has changed. (1 point)
Table 1

x
f(x)
0
−3
1
6
2
−9

Table 2

x
g(x)
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 need to replace \( x \) in \( f(x) \) with \( \frac{x}{k} \). This means we substitute \( x \) with \( 2x \) in the function \( f(x) \).

Thus, \( g(x) \) will be defined as:

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

Calculating \( g(x) \):

\[
g(x) = -12(4x^2) + 42x - 3
\]
\[
g(x) = -48x^2 + 42x - 3
\]

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

**Finding \( g(0) \)**:
\[
g(0) = -48(0)^2 + 42(0) - 3 = -3
\]

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

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

Now we can fill in Table 2:

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

So, the completed Table 2 is as follows:

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