Questions: Given the function below: f(x)=1.5(x-6)(x-4) What is the vertex for the graph of f(x) ? Enter your answer using coordinate pair notation. What is the stretch/compression factor for the graph of f(x) when compared to the graph of y=x^2 ? Stretch/Compression Factor:

Solution Steps

To find the vertex of the quadratic function \( f(x) = 1.5(x-6)(x-4) \), we can use the fact that the vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. Alternatively, we can find the vertex by averaging the roots and then substituting back into the function to find the y-coordinate.

To determine the stretch/compression factor, we compare the coefficient of the quadratic term in \( f(x) \) to the standard quadratic function \( y = x^2 \).

1. Identify the roots of the quadratic function.
2. Calculate the x-coordinate of the vertex by averaging the roots.
3. Substitute the x-coordinate back into the function to find the y-coordinate of the vertex.
4. The coefficient of the quadratic term in \( f(x) \) gives the stretch/compression factor.

The function \( f(x) = 1.5(x - 6)(x - 4) \) has roots at \( x = 4 \) and \( x = 6 \).

The x-coordinate of the vertex is calculated as the average of the roots: \[ x_{\text{vertex}} = \frac{4 + 6}{2} = 5 \] Next, we substitute \( x_{\text{vertex}} \) back into the function to find the y-coordinate: \[ y_{\text{vertex}} = f(5) = 1.5(5 - 6)(5 - 4) = 1.5(-1)(1) = -1.5 \] Thus, the vertex of the function is \( (5, -1.5) \).

The stretch/compression factor of the function compared to \( y = x^2 \) is given by the coefficient of the quadratic term in \( f(x) \), which is \( 1.5 \).

Final Answer