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Chapter 8: Problem 31

Graph the function, label the vertex, and draw the axis of symmetry. $$ f(x)=-2(x+5)^{2} $$

Short Answer

Expert verified
Vertex: (-5, 0). Axis of symmetry: x = -5. Parabola opens downward.

Step by step solution

01

Identify the Vertex

The given function is in the form y = a(x - h)^2 + k which is a parabola. First, rewrite the function in this standard form. Given function: f(x) = -2(x + 5)^2 which is of the form y = -2(x - (-5))^2 + 0. Therefore, the vertex (h, k) is at (-5, 0).
02

Determine the Axis of Symmetry

The axis of symmetry for a parabola of the form y = a(x - h)^2 + k is the vertical line through the vertex. For this function, the axis of symmetry is the vertical line x = -5.
03

Graph the Function

To graph f(x) = -2(x + 5)^2: 1. Plot the vertex (-5, 0). 2. Since the coefficient of (x + 5)^2 is negative, the parabola opens downward. 3. Plot additional points to get the shape of the parabola, such as calculating f(-4), f(-6), and so on. 4. Draw the axis of symmetry, which is the line x = -5. 5. Sketch the parabola passing through the vertex and symmetrically placed points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. For the function given in the exercise, f(x) = -2(x + 5)^2, the vertex forms the peak of this downward-opening parabola. To find it, we rewrite the function in the standard form y = a(x - h)^2 + k, where (h, k) is the vertex.

In our case, we have y = -2(x - (-5))^2 + 0, revealing that the vertex is at (-5, 0). This point represents the maximum value of the parabola because -2 (the coefficient of (x + 5)^2) is negative. When you plot the vertex on a graph, it helps guide the shape and position of the whole parabola.

Remember:
  • The vertex is the turning point of the parabola.
  • Negative coefficients mean the vertex is a maximum.
  • Positive coefficients mean the vertex is a minimum.
Axis of Symmetry
The axis of symmetry of a parabola is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For any quadratic function in the form of y = a(x - h)^2 + k, the equation of the axis of symmetry is x = h.

In our example function, f(x) = -2(x + 5)^2, rewriting it as y = -2(x - (-5))^2 + 0 shows that h = -5. Therefore, the axis of symmetry is the line x = -5. This line is crucial for graphing because it ensures the graph is symmetrical about the vertex.

Key points to remember:
  • The axis of symmetry is always a vertical line x = h.
  • It splits the parabola into two equal halves.
  • Helps identify symmetrical points on the graph.
Standard Form of a Quadratic Function
Quadratic functions are often written in the standard form y = a(x - h)^2 + k, where a, h, and k are constants. This form highlights key features, such as the vertex and the orientation of the parabola.

For the quadratic function in the given exercise, f(x) = -2(x + 5)^2, it is already nearly in the standard form. We adjust it to match y = a(x - h)^2 + k by writing y = -2(x - (-5))^2 + 0. Here, a = -2, h = -5, and k = 0.

Important aspects of the standard form:
  • a determines the direction and width of the parabola. Negative 'a' means it opens downward; positive 'a' means it opens upward.
  • (h, k) represents the vertex, the peak or trough of the parabola.
  • This form simplifies the process of identifying and graphing key points.

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