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Chapter 8: Problem 43

Sketch the graph of each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ f(x)=\frac{1}{4} x^{2}-9 $$

Short Answer

Expert verified
Vertex: (0, -9). Axis of symmetry: x = 0. Parabola opens upwards.

Step by step solution

01

Identify the Vertex

In the function \( f(x) = \frac{1}{4}x^2 - 9 \), the quadratic term is \( \frac{1}{4}x^2 \) and the constant term is \(-9\). The formula for a quadratic function is \( f(x) = ax^2 + bx + c \). For this function, \( a = \frac{1}{4} \), \( b = 0 \), and \( c = -9 \). The vertex form of a parabola is \( (h, k) \), where \( h = -\frac{b}{2a} \). Since \( b = 0 \), the vertex is at \( h = 0 \). Thus, the vertex is at \( (0, -9) \).
02

Determine the Axis of Symmetry

The axis of symmetry for a parabola given by a quadratic function \( ax^2 + bx + c \) is the vertical line \( x = h \), where \( h \) is the x-coordinate of the vertex. We found the vertex to be at \( (0, -9) \), so the axis of symmetry is \( x = 0 \).
03

Sketch the Graph

The parabola \( f(x) = \frac{1}{4}x^2 - 9 \) is a vertically oriented parabola that opens upwards because \( a = \frac{1}{4} \), which is positive. The vertex at \( (0, -9) \) is the lowest point on the graph. The axis of symmetry is the line \( x = 0 \). Draw a parabola with these features and ensure that it flattens as it moves away from the vertex because the coefficient \( \frac{1}{4} \) controls the width, making it narrower than \( x^2 \).

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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 a crucial point on the graph of a quadratic function. It represents the peak or the lowest point, depending on whether the parabola opens upwards or downwards. In the quadratic equation, such as \[ f(x) = \frac{1}{4}x^2 - 9, \]the vertex can be determined using the formula for finding the vertex from the standard form \[ ax^2 + bx + c. \] To locate the vertex, remember:
  • The formula to find the x-coordinate of the vertex is \( h = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the quadratic equation.
  • Since \( b = 0 \) in this function, the x-coordinate \( h \) is 0.
  • After finding \( h \), substitute it back into the function to find the y-coordinate \( k \). Here, \( k = f(0) = -9 \).
Thus, the vertex is at the point \( (0, -9) \). It represents the lowest point of this particular upward opening parabola.
Axis of Symmetry
The axis of symmetry is an imaginary vertical line that divides the parabola into two mirror-image halves. This line always passes through the vertex of the parabola. For any quadratic equation \( ax^2 + bx + c \), finding the axis of symmetry is straightforward.
  • The x-coordinate of the vertex, \( h \), also gives you the equation for the axis of symmetry.
  • In the example \( f(x) = \frac{1}{4}x^2 - 9 \), the vertex is at \( (0, -9) \). Therefore, the axis of symmetry is the line \( x = 0 \).
The axis of symmetry helps to visualize and graph the parabola efficiently. By knowing this line, you know that any point on one side of the parabola has a corresponding point on the other side.
Graphing Parabolas
Graphing a quadratic function involves sketching the parabola on a coordinate plane. It utilizes key characteristics such as the vertex, axis of symmetry, and the orientation (whether the parabola opens upwards or downwards). To graph the function \[ f(x) = \frac{1}{4}x^2 - 9, \]follow these steps:
  • Start by plotting the vertex \( (0, -9) \), as it is the lowest point in this upward-opening parabola.
  • Draw the axis of symmetry as a vertical line through the vertex, here it is \( x = 0 \).
  • Because \( a = \frac{1}{4} \) is positive, the parabola opens upwards. This coefficient also makes the parabola wider than one with \( a = 1 \), indicating it will rise more slowly.
  • To complete the sketch, choose additional points on one side of the vertex, calculate their corresponding \( y \)-values, and reflect these points over the axis of symmetry.
By following these steps, the parabola can be sketched accurately on the graph, helping you understand its shape and key features.

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Most popular questions from this chapter

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 . $$ f(x)=-4 x^{2}+8 x $$

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 8.1. $$ z^{2}-3 z $$

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 . $$ f(x)=2 x^{2}+4 x+5 $$

Solve by completing the square. See Section 8.1. $$ x^{2}+4 x=12 $$

Sketch the graph of each function. See Section 8.5. $$ f(x)=2(x-3)^{2}+2 $$
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