Question

sketch the graph of the quadratic function and the axis of symmetry.
$f(x) = -3(x + 2)^2 + 1$

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: Identify the vertex form

The quadratic function is given in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and the axis of symmetry is \( x = h \). In the function \( f(x) = -3(x + 2)^2 + 1 \), we can rewrite \( (x + 2) \) as \( (x - (-2)) \), so \( h = -2 \) and \( k = 1 \).

Step2: Determine the axis of symmetry

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

Step3: Analyze the parabola's direction and vertex

The coefficient \( a = -3 \) is negative, so the parabola opens downward. The vertex of the parabola is at \( (-2, 1) \) because \( (h, k) = (-2, 1) \). To sketch the graph, we can plot the vertex, use the axis of symmetry \( x = -2 \) (a dashed vertical line), and then find a few more points by plugging in values of \( x \) (e.g., \( x = -1 \), \( x = -3 \)) into the function to get corresponding \( y \)-values and plot those points, then draw the parabola (a solid curve) opening downward with the vertex at \( (-2, 1) \) and symmetric about \( x = -2 \).

Answer:

The axis of symmetry is the vertical line \( x = -2 \), and the graph of the quadratic function \( f(x) = -3(x + 2)^2 + 1 \) is a parabola opening downward with vertex at \( (-2, 1) \), symmetric about the line \( x = -2 \). (When using the graphing tool, graph the parabola as a solid curve and the line \( x = -2 \) as a dashed line.)