Question

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

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 the axis of symmetry is \( x = h \).

Step2: Identify \( h \) in given function

For the function \( f(x) = -3(x + 2)^2 + 4 \), we can rewrite \( (x + 2) \) as \( (x - (-2)) \). So, comparing with \( a(x - h)^2 + k \), we have \( h = -2 \) and \( k = 4 \).

Step3: Determine axis of symmetry

Using the formula for the axis of symmetry from the vertex form, since \( h = -2 \), the axis of symmetry is \( x = -2 \).

Step4: Analyze the parabola's shape

The coefficient \( a = -3 \), which is negative, so the parabola opens downward. The vertex is at \((-2, 4)\). To sketch the graph, plot the vertex, then use the fact that for quadratic functions, we can find other points (e.g., when \( x = -1 \), \( f(-1) = -3(-1 + 2)^2 + 4 = -3(1) + 4 = 1 \); when \( x = -3 \), \( f(-3) = -3(-3 + 2)^2 + 4 = -3(1) + 4 = 1 \); when \( x = 0 \), \( f(0) = -3(0 + 2)^2 + 4 = -3(4) + 4 = -8 \); when \( x = -4 \), \( f(-4) = -3(-4 + 2)^2 + 4 = -3(4) + 4 = -8 \)). Then draw a solid curve through these points (and others) opening downward, with the vertex at \((-2, 4)\), and draw a dashed line at \( x = -2 \) for the axis of symmetry.

Answer:

The axis of symmetry is the vertical line \( x = -2 \). The graph of \( f(x) = -3(x + 2)^2 + 4 \) is a parabola opening downward with vertex at \((-2, 4)\), and the axis of symmetry (dashed line) is \( x = -2 \). When sketching, plot the vertex, use symmetric points to draw the solid parabola, and draw \( x = -2 \) as a dashed line.