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Chapter 8: Problem 21

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples 6 and 7 . $$ h(x)=\frac{1}{3} x^{2} $$

Short Answer

Expert verified
Vertex: (0, 0), Axis of Symmetry: x = 0, Upward-opening parabola.

Step by step solution

01

Identify the Vertex

The vertex of a quadratic function in the form of \( ax^2 + bx + c \) is found at \( x = -\frac{b}{2a} \). For the function \( h(x) = \frac{1}{3}x^2 \), \( a = \frac{1}{3} \) and \( b = 0 \), hence the vertex is at \( x = 0 \). Evaluating \( h(0) \), we have \( h(0) = \frac{1}{3}(0)^2 = 0 \). Thus, the vertex is \((0, 0)\).
02

Determine the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at \((0, 0)\), the axis of symmetry is \( x = 0 \).
03

Plot the Vertex and Axis of Symmetry

On a coordinate plane, plot the point \((0, 0)\) which represents the vertex. Draw a dashed vertical line through \( x = 0 \) to represent the axis of symmetry.
04

Sketch the Parabola

The function \( h(x) = \frac{1}{3}x^2 \) is a parabola that opens upwards as \( a = \frac{1}{3} > 0 \). Since the coefficient \( a \) is less than 1, the parabola is wider than the standard \( x^2 \) parabola. Starting from the vertex at \((0, 0)\), plot a few points such as \((1, \frac{1}{3})\) and \((-1, \frac{1}{3})\), and mirror them over the axis of symmetry. Then, sketch a smooth curve through these points.

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Key Concepts

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

Vertex
In a quadratic function, the vertex represents the highest or lowest point on the graph, depending on the parabola's orientation. It's a key feature that gives the function's maximum or minimum value. For the function \( h(x) = \frac{1}{3}x^2 \), we find the vertex using the formula \( x = -\frac{b}{2a} \). In this case, \( a = \frac{1}{3} \) and \( b = 0 \), leading us to \( x = 0 \).

Once we have the \( x \)-value, we substitute it back into the function to find \( y \). So here, \( h(0) = \frac{1}{3}(0)^2 = 0 \), thus giving us the vertex at the point \( (0, 0) \).

The vertex is crucial:
  • It's where the parabola turns direction.
  • Indicates the axis of symmetry.
  • Helps in plotting the graph accurately.
Axis of Symmetry
The axis of symmetry in a quadratic function is an invisible vertical line that divides the parabola into two mirror-image halves. This line always passes through the vertex, effectively balancing the parabola on either side.

For our function \( h(x) = \frac{1}{3}x^2 \), since the vertex is at \( (0, 0) \), the axis of symmetry is the line \( x = 0 \), also known as the \( y \)-axis.

This characteristic line has several roles:
  • It ensures each corresponding point on the parabola has a mirror point.
  • Helps in verifying the accuracy of your parabola sketch.
  • Facilitates finding other points when graphing.
Parabola
A parabola is the U-shaped graph that is most commonly associated with quadratic functions. The form \( h(x) = ax^2 \) typically results in a graph that opens upwards when \( a > 0 \) or downwards when \( a < 0 \).

For \( h(x) = \frac{1}{3}x^2 \), since \( a = \frac{1}{3} > 0 \), the parabola opens upwards. The vertex \((0, 0)\) is at the lowest point because the parabola is upwards opening.

To sketch the parabola:
  • Start with plotting the vertex.
  • Determine a few additional points. For example, for \( x = 1 \), we find \( h(1) = \frac{1}{3}(1)^2 = \frac{1}{3} \).
  • Plot both points \( (1, \frac{1}{3}) \) and \( (-1, \frac{1}{3}) \) and draw a smooth curve through these, making sure it is symmetric about the axis.
The coefficient \( \frac{1}{3} \) affects the width, making this parabola wider compared to one shaped by \( x^2 \). The smaller the coefficient, the wider the graph.

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Most popular questions from this chapter

Without solving, determine whether the solutions of each equation are real numbers or complex but not real numbers. See the Concept Check in this section. $$ 3 z^{2}=10 $$

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 8.1. $$ y^{2}+y $$

Sketch the graph of each function. See Section 8.5. $$ f(x)=(x+5)^{2}+2 $$

Solve by completing the square. See Section 8.1. $$ z^{2}-8 z=2 $$

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}+1 $$
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