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 12
. 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

wrong

Answer 2

To find the values of \( g(x) \) based on the transformation of the function \( f(x) = -12x^2 + 21x - 3 \) through a horizontal stretch by a factor of \( k = 12 \), we first need to understand how the transformation affects the input values.

A horizontal stretch by a factor of \( k \) means that if we have a function \( f(x) \), the new function \( g(x) \) can be represented as:

\[
g(x) = f\left(\frac{x}{k}\right) = f\left(\frac{x}{12}\right)
\]

We will use this to calculate \( g(0) \), \( g(1) \), and \( g(2) \) based on the values of \( f(x) \) from Table 1:

### Calculating the values for \( g(x) \)

1. **For \( g(0) \)**:

\[
g(0) = f\left(\frac{0}{12}\right) = f(0)
\]

From Table 1:
\[
f(0) = -3
\]
So,
\[
g(0) = -3
\]

2. **For \( g(1) \)**:

\[
g(1) = f\left(\frac{1}{12}\right)
\]

To find \( f\left(\frac{1}{12}\right) \), we substitute \( x = \frac{1}{12} \) into the function \( f(x) \):

\[
f\left(\frac{1}{12}\right) = -12\left(\frac{1}{12}\right)^2 + 21\left(\frac{1}{12}\right) - 3
\]

Calculating each term:
\[
-12\left(\frac{1}{12}\right)^2 = -12\left(\frac{1}{144}\right) = -\frac{12}{144} = -\frac{1}{12}
\]

\[
21\left(\frac{1}{12}\right) = \frac{21}{12} = \frac{7}{4}
\]

So,
\[
f\left(\frac{1}{12}\right) = -\frac{1}{12} + \frac{7}{4} - 3
\]

Convert \(-3\) into twelfths:
\[
-3 = -\frac{36}{12}
\]

Now add these together:
\[
-\frac{1}{12} + \frac{21}{12} - \frac{36}{12} = \frac{-1 + 21 - 36}{12} = \frac{-16}{12} = -\frac{4}{3}
\]

Thus,
\[
g(1) = -\frac{4}{3}
\]

3. **For \( g(2) \)**:

\[
g(2) = f\left(\frac{2}{12}\right) = f\left(\frac{1}{6}\right)
\]

Substitute \( x = \frac{1}{6} \) into function \( f(x) \):

\[
f\left(\frac{1}{6}\right) = -12\left(\frac{1}{6}\right)^2 + 21\left(\frac{1}{6}\right) - 3
\]

Calculating each term:
\[
-12\left(\frac{1}{6}\right)^2 = -12\left(\frac{1}{36}\right) = -\frac{12}{36} = -\frac{1}{3}
\]

\[
21\left(\frac{1}{6}\right) = \frac{21}{6} = \frac{7}{2}
\]

So,
\[
f\left(\frac{1}{6}\right) = -\frac{1}{3} + \frac{21}{6} - 3
\]

Convert \(-3\) into sixths:
\[
-3 = -\frac{18}{6}
\]

Now combine:
\[
-\frac{1}{3} + \frac{21}{6} - \frac{18}{6} = -\frac{2}{6} + \frac{21}{6} - \frac{18}{6} = \frac{1}{6}
\]

Thus,
\[
g(2) = \frac{1}{6}
\]

### Finalizing Table 2

Now we can complete Table 2:

| x | g(x) |
|---|-----------|
| 0 | -3 |
| 1 | -\(\frac{4}{3}\) |
| 2 | \(\frac{1}{6}\) |

So the completed values for \( g(x) \) are:
- \( g(0) = -3 \)
- \( g(1) = -\frac{4}{3} \)
- \( g(2) = \frac{1}{6} \)