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

To horizontally stretch the function \( f(x) = -12x^2 + 21x - 3 \) by a factor of \( k = 12 \), we need to replace \( x \) in \( f(x) \) with \( \frac{x}{12} \). Therefore, the new function \( g(x) \) will be:

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

Now we simplify \( g(x) \):

\[
g(x) = -12 \cdot \frac{x^2}{144} + 21 \cdot \frac{x}{12} - 3
\]
\[
= -\frac{x^2}{12} + \frac{21x}{12} - 3
\]

Let's compute \( g(x) \) for the values in Table 2:

1. For \( x = 0 \):
\[
g(0) = -\frac{0^2}{12} + \frac{21 \cdot 0}{12} - 3 = -3.
\]

2. For \( x = 1 \):
\[
g(1) = -\frac{1^2}{12} + \frac{21 \cdot 1}{12} - 3 = -\frac{1}{12} + \frac{21}{12} - 3.
\]
\[
= \frac{20}{12} - 3 = \frac{20}{12} - \frac{36}{12} = -\frac{16}{12} = -\frac{4}{3}.
\]

3. For \( x = 2 \):
\[
g(2) = -\frac{2^2}{12} + \frac{21 \cdot 2}{12} - 3 = -\frac{4}{12} + \frac{42}{12} - 3.
\]
\[
= \frac{38}{12} - 3 = \frac{38}{12} - \frac{36}{12} = \frac{2}{12} = \frac{1}{6}.
\]

Let's summarize the results in Table 2:

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

So the completed table shows how the function \( g(x) \) has changed with the given inputs.