Question

What is true about the function h(x) = x2 + 20x – 17? Check all that apply.

The vertex of h is (–10, –117).
The vertex form of the function is h(x) = (x + 20)2 – 17.
The maximum value of the function is –17.
To graph the function h, shift the graph of f(x) = x2 left 10 units and down 117 units.
The axis of symmetry of function h is x = 20.

Answer 1

To analyze the function \( h(x) = x^2 + 20x - 17 \), we will check each statement provided.

1. **The vertex of h is (–10, –117)**.

To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \).
Here, \( a = 1 \) and \( b = 20 \):
\[
x = -\frac{20}{2 \cdot 1} = -10.
\]
Now we substitute \( x = -10 \) back into the function to find the y-coordinate:
\[
h(-10) = (-10)^2 + 20(-10) - 17 = 100 - 200 - 17 = -117.
\]
Therefore, the vertex is indeed \( (-10, -117) \), so this statement is **True**.

2. **The vertex form of the function is h(x) = (x + 20)² – 17**.

The correct vertex form of a quadratic function is \( h(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. We already found the vertex to be \( (-10, -117) \). To rewrite \( h(x) \) in vertex form, complete the square:
\[
h(x) = x^2 + 20x - 17.
\]
Factor out the quadratic and linear parts:
\[
h(x) = (x^2 + 20x + 100) - 100 - 17 = (x + 10)^2 - 117.
\]
Thus, the vertex form is \( h(x) = (x + 10)^2 - 117 \), not \( (x + 20)^2 - 17 \). So this statement is **False**.

3. **The maximum value of the function is –17.**

Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), there is a minimum value at the vertex, which we found to be \( -117 \). Thus, the function does not have a maximum value, and this statement is **False**.

4. **To graph the function h, shift the graph of f(x) = x² left 10 units and down 117 units.**

Since the vertex is \( (-10, -117) \) and the function opens upwards, the transformation involves shifting the base graph \( f(x) = x^2 \) left 10 units to reach \( (0, 0) \) to \( (-10, 0) \) and then down 117 units to reach \( (-10, -117) \). This statement is **True**.

5. **The axis of symmetry of function h is x = 20.**

The axis of symmetry is given by \( x = -10 \) because the vertex is at \( (-10, -117) \). Thus, this statement is **False**.

In summary, the true statements about the function \( h(x) = x^2 + 20x - 17 \) are:

- The vertex of h is (–10, –117).
- To graph the function h, shift the graph of f(x) = x² left 10 units and down 117 units.