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) = x² - 9\
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 Quadratic Function Form

The general form of a quadratic function is \( h(x) = ax^2 + bx + c \). For \( h(x)=x^2 - 9 \), we have \( a = 1 \), \( b = 0 \), \( c=-9 \). The vertex form of a quadratic is \( h(x)=a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h(x)=(x - 0)^2 - 9 \), so the vertex is \((0, -9)\).

Step2: Find Axis of Symmetry

The formula for the axis of symmetry of a quadratic \( ax^2+bx + c \) is \( x = -\frac{b}{2a} \). Substituting \( a = 1 \), \( b = 0 \), we get \( x = -\frac{0}{2(1)} = 0 \). So the axis of symmetry is the line \( x = 0 \) (the y - axis).

Step3: Sketching the Graph

• The parabola opens upwards because \( a = 1>0 \).
• The vertex is at \((0, -9)\), so it intersects the y - axis at \((0, -9)\).
• To find the x - intercepts, set \( h(x)=0 \), so \( x^2 - 9 = 0 \), which factors as \((x - 3)(x + 3)=0\). Thus, the x - intercepts are at \( x = 3 \) and \( x=-3 \), so the points \((3,0)\) and \((-3,0)\) are on the graph.
• The axis of symmetry is the vertical line \( x = 0 \), which is a dashed line through the vertex, parallel to 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 \( x = 0 \). The graph is a parabola opening upwards with vertex at \((0, -9)\), x - intercepts at \((3,0)\) and \((-3,0)\), and the axis of symmetry is the line \( x = 0 \) (the y - axis, drawn as a dashed line).