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Chapter 11: Problem 44

For each of the following, graph the function, label the vertex, and draw the axis of symmetry. $$ h(x)=-2\left(x+\frac{1}{2}\right)^{2} $$

Short Answer

Expert verified
The vertex is \[ \bigg(-\frac{1}{2}, 0\bigg) \]. The axis of symmetry is \[ x = -\frac{1}{2} \].

Step by step solution

01

Identify the Vertex

The given function is in vertex form: \[ h(x) = -2\bigg(x + \frac{1}{2}\bigg)^{2} \] In this form \[ h(x) = a \bigg(x - h \bigg)^{2} + k \] the vertex \[ (h, k) \] is given directly. Here, \[h = -\frac{1}{2} \text{ and } k = 0 \] Thus, the vertex is \[ \bigg(-\frac{1}{2}, 0 \bigg) \]
02

Determine the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is the vertical line that passes through the vertex. Therefore, for the function \[ h(x)=-2\bigg(x+\frac{1}{2}\bigg)^{2} \] the axis of symmetry is \[ x = -\frac{1}{2} \]
03

Choose Points to Plot the Graph

To plot the graph, choose a few values of \[ x \] and calculate the corresponding \[ h(x) \]. For example, when \[ x = 0 \] \[ h(0) = -2 \bigg(0 + \frac{1}{2}\bigg)^{2} = -2 \bigg(\frac{1}{2}\bigg)^{2} = -2 \bigg(\frac{1}{4}\bigg) = -\frac{1}{2} \] Similarly, choose other values such as \[ x = -1 \] and \[ x = 1 \]. Calculate the respective \[ h(x) \] values and plot them on the graph.
04

Plot and Draw the Graph

With the vertex \[ \bigg(-\frac{1}{2}, 0\bigg) \] and the chosen points from Step 3, plot these on a coordinate system. Draw a smooth curve through these points to form the parabola. Ensure that the parabola opens downwards because the coefficient of \[ \bigg(x + \frac{1}{2}\bigg)^{2} \] is negative.
05

Label the Graph

Label the vertex \[ \bigg(-\frac{1}{2}, 0\bigg) \] and clearly draw and label the axis of symmetry \[ x = -\frac{1}{2} \]. Ensure the graph reflects the precise shape of the parabola.

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Key Concepts

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

vertex form of quadratic function
Understanding the vertex form of a quadratic function is crucial for graphing parabolas. The vertex form is given by

\[ f(x) = a(x - h)^2 + k \] where \[ a, h, \text{ and } k \] are constants. Here:
  • \[a \] determines the direction and width of the parabola. If \[a \] is negative, the parabola opens downwards.
  • \[h \] is the x-coordinate of the vertex.
  • \[k \] is the y-coordinate of the vertex.

    By identifying the values of \[h \] and \[k \], you can directly find the vertex of the parabola. In the given example, \[ h(x) = -2\bigg(x+\frac{1}{2}\bigg)^{2} \], the equation can be rewritten in the vertex form with \[ a = -2 \], \[ h = -\frac{1}{2} \], and \[ k = 0 \].
vertex identification
Identifying the vertex is simple once you have the equation in vertex form. The vertex is the point where the parabola changes direction. It is represented as \[(h, k)\]. In our case, \[ h = -\frac{1}{2} \] and \[ k = 0 \] giving us the vertex point \[ \bigg(-\frac{1}{2}, 0\bigg) \]. This vertex point can be used as a starting point for plotting the parabola. Mark this exact point on the graph before plotting other points.
axis of symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images. It always passes through the vertex. For a given quadratic function in the vertex form \[ f(x) = a(x - h)^2 + k \], the axis of symmetry is the line \[ x = h \].

In our example, the vertex is at \[-\frac{1}{2} \]. So the axis of symmetry will be the line \[ x = -\frac{1}{2} \]. This line helps you verify that the points on either side of the vertex are symmetric. Draw this line on your graph and label it clearly.
plotting points on a graph
Plotting points involves selecting various \[ x \] values and computing their corresponding \[ y \] values. Start by choosing \[ x \] values close to the vertex. Calculate \[ h(x) \] for these values:
  • For instance, if \[ x = 0 \], \[ h(0) = -2 \bigg(0 + \frac{1}{2}\bigg)^2 = -\frac{1}{2} \].
  • When \[ x = -1 \], calculate \[ h(-1) \].
  • Continue this for other points like \[ x = 1 \]. Calculate the \[ y \]-values and plot these points on the graph.
    Connect these points smoothly. The parabola's shape starts forming. Ensure the curve opens downward since the coefficient \[ a \] is negative (\

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