To create the function \( g(x) \) from the parent function \( f(x) = -12x^2 + 21x - 3 \) by horizontally stretching it by a factor of \( k = \frac{1}{2} \), we replace \( x \) with \( \frac{x}{k} \) in the function \( f(x) \).
In this case, since \( k = \frac{1}{2} \):
\[ g(x) = f\left(\frac{x}{\frac{1}{2}}\right) = f(2x) \]
Now we will compute \( g(x) \) for the x-values provided in Table 2: 0, 1, and 2.
Calculate
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For \( x = 0 \): \[ g(0) = f(2 \times 0) = f(0) = -3 \]
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For \( x = 1 \): \[ g(1) = f(2 \times 1) = f(2) \] We need to compute \( f(2) \): \[ f(2) = -12(2^2) + 21(2) - 3 = -12(4) + 42 - 3 = -48 + 42 - 3 = -9 \] Thus, \( g(1) = -9 \).
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For \( x = 2 \): \[ g(2) = f(2 \times 2) = f(4) \] We compute \( f(4) \): \[ f(4) = -12(4^2) + 21(4) - 3 = -12(16) + 84 - 3 = -192 + 84 - 3 = -111 \] Thus, \( g(2) = -111 \).
Summary of Results
The values for \( g(x) \) are as follows:
| Table 2: x-values | g(x) values | |--------------------|--------------| | 0 | -3 | | 1 | -9 | | 2 | -111 |
So the completed table would look like this:
| x-value | g(x) | |---------|------| | 0 | -3 | | 1 | -9 | | 2 | -111 |
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