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 |