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 + 1)^2 + 1 )
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 \( h \) and \( k \) in \( h(x)=(x + 1)^2+1 \)

Rewrite \( (x + 1)^2 \) as \( (x - (-1))^2 \). So comparing with \( a(x - h)^2 + k \), we have \( h=-1 \) and \( k = 1 \).

Step3: Determine vertex and axis of symmetry

The vertex is \((-1, 1)\) because \( h=-1 \) and \( k = 1 \). The axis of symmetry is the vertical line \( x=-1 \) (since for vertex form \( x = h \)).

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

Answer:

Vertex: \((-1, 1)\)
Axis of symmetry: \( x = -1 \)
(For the graph, plot the vertex \((-1, 1)\), points like \((0, 2)\), \((-2, 2)\), draw the upward - opening parabola, and the dashed line \( x=-1 \))