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 + 5)^2\\)\\(\\)\\(\\)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 vertex form of quadratic

The vertex form of a quadratic function is \( h(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and the axis of symmetry is \( x = h \).

Step2: Identify \( h \) in given function

For \( h(x) = (x + 5)^2 \), we can rewrite it as \( h(x) = 1(x - (-5))^2 + 0 \). So, the vertex is \((-5, 0)\) and the axis of symmetry is \( x = -5 \).
To sketch the graph: The parabola opens upwards (since \( a = 1>0 \)). Plot the vertex \((-5, 0)\). Then, we can find other points. For example, when \( x = -4 \), \( h(-4)=(-4 + 5)^2=1 \), so the point \((-4, 1)\) is on the parabola. When \( x = -6 \), \( h(-6)=(-6 + 5)^2 = 1 \), so \((-6, 1)\) is also on the parabola. Draw a smooth curve through these points (solid curve) and draw the dashed line \( x=-5 \) for the axis of symmetry.

Answer:

Vertex: \((-5, 0)\)
Axis of symmetry: \( x = -5 \)
(For graphing: Plot vertex \((-5,0)\), use points like \((-4,1)\), \((-6,1)\) to sketch the upward - opening parabola (solid curve) and draw \( x = -5 \) as a dashed line.)