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

Question

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:

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