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

Question

Complete the table to show how the function has changed.
Table 1: x:0, 1, 2 f(x):-3, 6, -9 Table 2: x: 0, 1, 2 g(x):

Answer 1

To find the new function \( g(x) \) obtained by horizontally stretching the function \( f(x) = -12x^2 + 21x - 3 \) with a stretch factor \( k = \frac{1}{2} \), we follow these steps:

1. Identify the transformation: A horizontal stretch by a factor of \( k \) means we replace \( x \) with \( \frac{x}{k} \) in the original function. So for our function \( g(x) \), we have: \[ g(x) = f\left(\frac{x}{\frac{1}{2}}\right) = f(2x) \]

2. Substitute to create \( g(x) \): Now, we can calculate \( g(x) \) using \( f(2x) \): \[ g(x) = -12(2x)^2 + 21(2x) - 3 \] Simplifying, we get: \[ g(x) = -12 \cdot 4x^2 + 42x - 3 = -48x^2 + 42x - 3 \]

3. Evaluate \( g(x) \) at the specified points: We need to evaluate \( g(x) \) at \( x = 0, 1, \text{ and } 2 \).

   • For \( x = 0 \): \[ g(0) = -48(0)^2 + 42(0) - 3 = -3 \]

   • For \( x = 1 \): \[ g(1) = -48(1)^2 + 42(1) - 3 = -48 + 42 - 3 = -9 \]

   • For \( x = 2 \): \[ g(2) = -48(2)^2 + 42(2) - 3 = -48 \cdot 4 + 84 - 3 = -192 + 84 - 3 = -111 \]

4. Fill in Table 2 with calculated values:

   | x | g(x) | |---|-------| | 0 | -3 | | 1 | -9 | | 2 | -111 |

So the final filled Table 2 is:

| x | g(x) | |----|-------| | 0 | -3 | | 1 | -9 | | 2 | -111 |