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Chapter 5: Problem 108

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) For which is the line of symmetry \(x=0 ?\)

Short Answer

Expert verified
a, b, c, d

Step by step solution

01

Identify the Standard Form of a Quadratic Function

The general form of a quadratic function is given by \( f(x) = ax^2 + bx + c \) or in the vertex form \( f(x) = a(x-h)^2 + k \) where \( h \) represents the x-coordinate of the vertex. The line of symmetry can be found from this value \( h \). If \( h = 0 \), then the line of symmetry is \( x = 0 \).
02

Extract the Vertex from Each Function

Determine \( h \) for each of the given functions:(a) \( f(x) = 2x^2 \)(b) \( f(x) = -x^2 \)(c) \( f(x) = \frac{1}{4}x^2 \)(d) \( f(x) = -5x^2 + 3 \)(e) \( f(x) = \frac{2}{3}(x-1)^2 - 3 \)(f) \( f(x) = -2(x+3)^2 + 1 \)(g) \( f(x) = (x-3)^2 + 1 \)(h) \( f(x) = -4(x+1)^2 -3 \). In forms that already display the vertex form, identify the value of \( h \).
03

Determine the Line of Symmetry

Based on the values of \( h \), the line of symmetry is \( x = h \). Find which functions have \( h = 0 \):(a) \( f(x) = 2x^2 \) 鈥� Here \( h = 0 \)(b) \( f(x) = -x^2 \) 鈥� Here \( h = 0 \)(c) \( f(x) = \frac{1}{4}x^2 \) 鈥� Here \( h = 0 \)(d) \( f(x) = -5x^2 + 3 \) 鈥� Here \( h = 0 \)(e) \( f(x) = \frac{2}{3}(x-1)^2 - 3 \) 鈥� Here \( h = 1 \)(f) \( f(x) = -2(x+3)^2 + 1 \) 鈥� Here \( h = -3 \)(g) \( f(x) = (x-3)^2 + 1 \) 鈥� Here \( h = 3 \)(h) \( f(x) = -4(x+1)^2 -3 \) 鈥� Here \( h = -1 \). Thus, the functions that have the line of symmetry \( x = 0 \) are \(a, b, c, d\).

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Key Concepts

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

Line of Symmetry
Understanding the line of symmetry in quadratic functions is important. It's the vertical line that divides the parabola into two mirror images. For a quadratic function in the form \( f(x) = a(x-h)^2 + k \), the line of symmetry is given by the equation \( x = h \).
This means that at \( h \), the function is symmetric. Each point on one side of this line has a corresponding point on the other side, equidistant from the line of symmetry.
Standard Form of a Quadratic Function
The standard form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \). Here are the meanings of the different components:
  • \( a \): The coefficient of \(x^2\), dictating the parabola鈥檚 direction and width.
  • \( b \): The coefficient of \(x\), which affects the position of the vertex horizontally.
  • \( c \): The constant term, which moves the parabola up or down vertically.
The standard form is useful for identifying the line of symmetry, vertex, and other properties of parabolas.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \). This form is particularly useful because it allows us to easily identify the vertex of the parabola.
  • \(h\): The x-coordinate of the vertex.
  • \(k\): The y-coordinate of the vertex.
The values \(h\) and \(k\) tell us the vertex鈥檚 location, shifting the parabola horizontally by \(h\) units and vertically by \(k\) units. Vertex form clearly shows the line of symmetry as \(x = h\).
Identifying Vertices
Identifying vertices in quadratic functions helps to understand their maximum or minimum points. Given a quadratic function in vertex form \( f(x) = a(x-h)^2 + k \), the vertex is directly visible as \((h, k)\).
For standard form \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). Once the x-coordinate is found, substitute it back into the function to find the y-coordinate, giving the vertex as \( \bigg( -\frac{b}{2a}, f\bigg(-\frac{b}{2a}\bigg) \bigg )\).
This process is crucial for sketching the graph and understanding the function's behavior.

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Most popular questions from this chapter

Charitable Giving. Over the last four decades, the amount of charitable giving in the United States has grown exponentially from approximately \(\$ 20.7\) billion in 1969 to approximately \(\$ 316.2\) billion in 2012 (Sources: Giving USA Foundation; Volunteering in America by the Corporation for National \& Community Service; National Philanthropic Trust; School of Philanthropy, Indiana University Purdue University Indianapolis). The exponential function $$ G(x)=20.7(1.066)^{x} $$ where \(x\) is the number of years after \(1969,\) can be used to estimate the amount of charitable giving, in billions of dollars, in a given year. Find the amount of charitable giving in \(1982,\) in \(1995,\) and in \(2010 .\) Then use this function to estimate the amount of charitable giving in \(2017 .\) Round to the nearest billion dollars. (IMAGE CANT COPY)

Using only a graphing calculator, determine whether the functions are inverses of each other. Consider the function \(f\) given by $$ f(x)=\left\\{\begin{array}{ll} x^{3}+2, & x \leq-1 \\ x^{2}, & -1

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Solve using any method. Given that \(a=\left(\log _{125} 5\right)^{\log _{5} 125},\) find the value of \(\log _{3} a\)
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