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.
$f(x) = (x - 6)^2$
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 \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \( x = h \) is the axis of symmetry.

Step2: Identify \( h \) and \( k \)

For \( f(x) = (x - 6)^2 \), we can rewrite it as \( f(x) = 1(x - 6)^2 + 0 \). So, \( h = 6 \) and \( k = 0 \).

Step3: Determine vertex and axis

The vertex is \((h, k) = (6, 0)\). The axis of symmetry is \( x = h \), so \( x = 6 \).

To sketch the graph: The parabola opens upward (since \( a = 1>0 \)) with vertex at \((6, 0)\). Plot the vertex, then find other points (e.g., when \( x = 5 \), \( f(5)=(5 - 6)^2 = 1 \); when \( x = 7 \), \( f(7)=(7 - 6)^2 = 1 \)), and draw the parabola. The axis of symmetry \( x = 6 \) is a dashed vertical line through \( x = 6 \).

Answer:

Vertex: \((6, 0)\)
Axis of symmetry: \( x = 6 \)