Question

sketch the graph of the quadratic function and the axis of symmetry. state the vertex, and give the equation for the axis of symmetry.\
\\( h(x) = x^2 + 9 \\)\
\\( \quad \\)\
use the graphing tool to graph the function as a solid curve and the axis of symmetry as a dashed line.\
click to enlarge graph

Explanation:

Step1: Recall the vertex form of a quadratic function

The vertex form of a quadratic function is \( h(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the given function \( h(x) = x^2 + 9 \), we can rewrite it as \( h(x) = 1(x - 0)^2 + 9 \).

Step2: Identify the vertex

From the vertex form \( h(x) = 1(x - 0)^2 + 9 \), we can see that \( h = 0 \) and \( k = 9 \). So the vertex of the parabola is \((0, 9)\).

Step3: Determine the axis of symmetry

For a quadratic function in the form \( h(x) = a(x - h)^2 + k \), the axis of symmetry is the vertical line \( x = h \). Since \( h = 0 \) in our function, the axis of symmetry is \( x = 0 \) (which is the y - axis).

Answer:

The vertex of the quadratic function \( h(x)=x^{2}+9 \) is \((0, 9)\) and the equation of the axis of symmetry is \( x = 0 \).