To create the function \( g(x) \) from the function \( f(x) = -12x^2 + 21x - 3 \) by horizontally stretching it with a factor of \( k = 12 \), we need to adjust the input of the function \( f \).
The transformation for a horizontal stretch by a factor of \( k \) is given by: \[ g(x) = f\left(\frac{x}{k}\right) \] In this case, \( k = 12 \), so: \[ g(x) = f\left(\frac{x}{12}\right) \]
Now we will compute the values of \( g(x) \) based on the values from the table of \( f(x) \).
Step 1: Determine \( g(0) \)
For \( x = 0 \): \[ g(0) = f\left(\frac{0}{12}\right) = f(0) = -3 \]
Step 2: Determine \( g(1) \)
For \( x = 1 \): \[ g(1) = f\left(\frac{1}{12}\right) \]
First, calculate \( f\left(\frac{1}{12}\right) \): \[ f\left(\frac{1}{12}\right) = -12\left(\frac{1}{12}\right)^2 + 21\left(\frac{1}{12}\right) - 3 \] \[ = -12\left(\frac{1}{144}\right) + 21\left(\frac{1}{12}\right) - 3 \] \[ = -\frac{1}{12} + \frac{21}{12} - 3 \] \[ = \frac{20}{12} - 3 = \frac{20}{12} - \frac{36}{12} = -\frac{16}{12} = -\frac{4}{3} \] Thus, \( g(1) = -\frac{4}{3} \).
Step 3: Determine \( g(2) \)
For \( x = 2 \): \[ g(2) = f\left(\frac{2}{12}\right) = f\left(\frac{1}{6}\right) \]
Next, compute \( f\left(\frac{1}{6}\right) \): \[ f\left(\frac{1}{6}\right) = -12\left(\frac{1}{6}\right)^2 + 21\left(\frac{1}{6}\right) - 3 \] \[ = -12\left(\frac{1}{36}\right) + 21\left(\frac{1}{6}\right) - 3 \] \[ = -\frac{1}{3} + \frac{21}{6} - 3 \] \[ = -\frac{1}{3} + \frac{21}{6} - \frac{18}{6} \] \[ = -\frac{1}{3} + \frac{3}{6} = -\frac{1}{3} + \frac{1}{2} \] Convert \( -\frac{1}{3} \) to sixths: \[ = -\frac{2}{6} + \frac{3}{6} = \frac{1}{6} \] Thus, \( g(2) = \frac{1}{6} \).
Final table values for \( g(x) \):
- \( g(0) = -3 \)
- \( g(1) = -\frac{4}{3} \)
- \( g(2) = \frac{1}{6} \)
So, the completed Table 2 looks like this:
| x | g(x) | |---|------------------| | 0 | -3 | | 1 | -\(\frac{4}{3}\) | | 2 | \(\frac{1}{6}\) |
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