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:
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For \( x = 0 \): \[ g(0) = -\frac{0^2}{12} + \frac{21 \cdot 0}{12} - 3 = -3. \]
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For \( x = 1 \): \[ g(1) = -\frac{1^2}{12} + \frac{21 \cdot 1}{12} - 3 = -\frac{1}{12} + \frac{21}{12} - 3. \] \[ = \frac{20}{12} - 3 = \frac{20}{12} - \frac{36}{12} = -\frac{16}{12} = -\frac{4}{3}. \]
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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.