Question

sketch the graph of the quadratic function and the axis of symmetry.
$f(x) = -3(x + 4)^2 + 2$

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: 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 \( x = h \). For \( f(x) = -3(x + 4)^2 + 2 \), we can rewrite \( (x + 4) \) as \( (x - (-4)) \), so \( h = -4 \) and \( k = 2 \).

Step2: Determine the axis of symmetry

From the vertex form, the axis of symmetry is \( x = h \). Since \( h = -4 \), the axis of symmetry is the vertical line \( x = -4 \).

Step3: Analyze the parabola's direction and vertex

The coefficient \( a = -3 \) is negative, so the parabola opens downward. The vertex is at \( (-4, 2) \). To sketch the graph, plot the vertex at \( (-4, 2) \), use the axis of symmetry \( x = -4 \) (dashed line) to find symmetric points. For example, when \( x = -3 \), \( f(-3) = -3(-3 + 4)^2 + 2 = -3(1)^2 + 2 = -1 \), so the point \( (-3, -1) \) is on the parabola. Its symmetric point across \( x = -4 \) is \( (-5, -1) \). Plot a few more points (e.g., \( x = -2 \): \( f(-2) = -3(-2 + 4)^2 + 2 = -3(4) + 2 = -10 \), symmetric point \( (-6, -10) \)) and draw the parabola as a solid curve opening downward with vertex at \( (-4, 2) \), and draw the dashed line \( x = -4 \) for the axis of symmetry.

Answer:

The axis of symmetry is \( x = -4 \). To sketch the graph: plot the vertex \((-4, 2)\), draw the parabola opening downward (using symmetric points) as a solid curve, and draw the dashed line \( x = -4 \) for the axis of symmetry.