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Section9.2Properties of Quadratic Functions

Figure9.2.1Alternative Video Lesson

Subsection9.2.1Introduction

A quadratic function has the form \(y=ax^2+bx+c\) where \(a, b\text{,}\) and \(c\) are real numbers with \(a \neq 0\text{.}\) The graph of a quadratic function is often referred to as a parabola.

As far as formulas are concerned, the key distinction between a quadratic function and a linear function is that a quadratic function has \(x^2\) in the highest-degreed term, whereas a linear function has just \(x\) in the highest-degreed term. In this section, we'll explore the numerical and graphical aspects of quadratic functions.

Subsection9.2.2Graphing Quadratic Functions Using a Table

The simpliest quadratic function is given by \(y=x^2\text{.}\) We'll start by creating a table of values and graphing this function:

\(x\) \(y=x^2\) Point
\(-3\) \((-3)^2=9\) \((-3,9)\)
\(-2\) \((-2)^2=4\) \((-2,4)\)
\(-1\) \((-1)^2=1\) \((-1,1)\)
\(0\) \(0^2=0\) \((0,0)\)
\(1\) \(1^2=1\) \((0,1)\)
\(2\) \(2^2=4\) \((2,4)\)
\(3\) \(3^2=9\) \((3,9)\)

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Table9.2.2Function values and points for \(y=x^2\)
Figure9.2.3Graph of \(y=x^2\)

There are a couple of key things to note in Table 9.2.2 and Figure 9.2.3. To start, the output of \(x^2\) makes all outputs positive, and results in a minimum value of \(0\text{.}\) In fact, all quadratic functions will have either a maximum or minimum value and have a similar shape as \(y=x^2\text{.}\) Also note that the function is symmetric about the place where its minimum value occurs. Lastly, the consecutive \(y\)-values do not increase by a constant amount in the way that linear functions do.

Example9.2.4

Create a table of values for the quadratic function \(y=x^2-4x+1\) and use this table to create its graph.

Solution

We'll start by choosing \(x\)-values between \(-3\) and \(3\text{:}\)

\(x\) \(y=x^2-4x+1\) Point
\(-3\) \((-3)^2-4(-3)+1=22\) \((-3,22)\)
\(-2\) \((-2)^2-4(-2)+1=13\) \((-2,4)\)
\(-1\) \((-1)^2-4(-1)+1=6\) \((-1,6)\)
\(0\) \(0^2-4(0)+1=1\) \((0,1)\)
\(1\) \(1^2-4(1)+1=-2\) \((0,1)\)
\(2\) \(2^2-4(2)+1=-3\) \((2,-3)\)
\(3\) \(3^2-4(3)+1=-2\) \((3,-2)\)

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Table9.2.5Function values and points for \(y=x^2-4x+1\)
Figure9.2.6Initial points for \(y=x^2-4x+1\)

This is a good start, but doesn't give us the whole picture. Based on the pattern we see in Table 9.2.5, we can see that the \(y\)-values will continue to increase. It will be helpful to continue our table past values of \(x=3\text{:}\)

\(x\) \(y=x^2-4x+1\) Point
\(-3\) \((-3)^2-4(-3)+1=22\) \((-3,22)\)
\(-2\) \((-2)^2-4(-2)+1=13\) \((-2,4)\)
\(-1\) \((-1)^2-4(-1)+1=6\) \((-1,6)\)
\(0\) \(0^2-4(0)+1=1\) \((0,1)\)
\(1\) \(1^2-4(1)+1=-2\) \((1,-2)\)
\(2\) \(2^2-4(2)+1=-3\) \((2,-3)\)
\(3\) \(3^2-4(3)+1=-2\) \((3,-2)\)
\(4\) \(4^2-4(4)+1=1\) \((4,1)\)
\(5\) \(5^2-4(5)+1=6\) \((5,6)\)

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Example9.2.7

Create a table of values for the quadratic function \(y=-\frac{1}{2}\left(x+1\right)^2+4\) and use this table to create its graph.

Solution

We'll start by choosing \(x\)-values between \(-5\) and \(5\text{,}\) and then we'll plot these and connect them with a curved line:

\(x\) \(y=-\frac{1}{2}(x+1)^2+4\) Point
\(-5\) \(-\frac{1}{2}(-5+1)^2+4=-4\) \((-5,-4)\)
\(-4\) \(-\frac{1}{2}(-4+1)^2+4=-0.5\) \((-4,-0.5)\)
\(-3\) \(-\frac{1}{2}(-3+1)^2+4=2\) \((-3,2)\)
\(-2\) \(-\frac{1}{2}(-2+1)^2+4=3.5\) \((-2,3.5)\)
\(-1\) \(-\frac{1}{2}(-1+1)^2+4=4\) \((-1,4)\)
\(0\) \(-\frac{1}{2}(0+1)^2+4=3.5\) \((0,3.5)\)
\(1\) \(-\frac{1}{2}(1+1)^2+4=2\) \((1,2)\)
\(2\) \(-\frac{1}{2}(2+1)^2+4=-0.5\) \((2,-0.5)\)
\(3\) \(-\frac{1}{2}(3+1)^2+4=-4\) \((3,-4)\)
\(4\) \(-\frac{1}{2}(4+1)^2+4=-8.5\) \((4,-8.5)\)
\(5\) \(-\frac{1}{2}(5+1)^2+4=-14\) \((5,-14)\)

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Table9.2.8Function values and points for \(y=-\frac{1}{2}\left(x+1\right)^2+4\)
Figure9.2.9Graph of \(y=-\frac{1}{2}\left(x+1\right)^2+4\)

Notice that in drawing the graph, a couple of points were excluded. We focused the middle of the graph, which was found to be \((-1,4)\text{,}\) as well as plenty of points around this.
Remark9.2.10

Notice that in Example 9.2.7, the function was not explicitly given in the form \(y=ax^2+bx+c\text{.}\) However, it could be written this way if we expanded the expression as follows:

\begin{align*} y&=-\frac{1}{2}\left(x+1\right)^2+4\\ y&=-\frac{1}{2}\left(x^2+2x+1\right)+4\\ y&=-\frac{1}{2}x^2-x-\frac{1}{2}+4\\ y&=-\frac{1}{2}x^2-x+\frac{7}{2} \end{align*}

Subsection9.2.3Properties of Quadratic Functions

Example9.2.11

A quadratic function given by \(y=-x^2+2x+8\) is graphed in Figure 9.2.12, which shows key features of quadratic functions. Notice that this parabola points down. Whether a parabola points up or down is determined by the sign of the coefficient on the \(x^2\) term.

The maximum (or minimum) point on the graph of a parabola is referred to as the vertex. The vertical line that passes through this point is referred to as its axis of symmetry.

Other key points on a graph are where \(x=0\) and where \(y=0\text{.}\) Just like with linear functions, the place(s) where the function hits the \(x\)-axis (i.e. where \(y=0\)) are referred to as \(x\)-intercepts. Similarly, the place where the function hits the \(y\)-axis is referred to as a \(y\)-intercept.

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Figure9.2.12Graph of a quadratic function showing key features

In Figure 9.2.12, the vertex is the point \((1,9)\) and the axis of symmetry is therefore the line \(x=1\text{.}\) The \(x\)-intercepts are \((-2,0)\) and \((4,0)\text{,}\) and the \(y\)-intercept is \((0,8)\text{.}\)

Vertex

The vertex of a parabola is the maximum or minimum point on the graph.

Axis of Symmetry

The axis of symmetry is the vertical line through which the vertex passes.

Horizontal Intercepts

The horizontal intercept(s) on the graph of a function occur where a function intersects the horizontal axis. The graph of a parabola can have \(0, 1,\) or \(2\) intercepts.

If the axes are labeled \(x\) and \(y\text{,}\) then the horizontal axis is the \(x\)-axis and the horizontal intercepts are the \(x\)-intercepts.

Vertical Intercept

The vertical intercept on the graph of a function occurs where the function intersects the vertical axis. Every parabola has exactly one vertical intercept.

If the axes are labeled \(x\) and \(y\text{,}\) then the vertical axis is the \(y\)-axis and the vertical intercept is the \(y\)-intercept.

List9.2.13Summary of Properties of Quadratic Functions
Exercise9.2.14

Use Figure 9.2.15 to answer the following questions.

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Figure9.2.15
Remark9.2.16

Although the outputs for \(y=x^2\) do not increase by a constant amount, the second differences do, as shown in Table 9.2.17.

\(x\) \(x^2\)
\(-3\) \(9\)
\(\stackrel{\nwarrow}{\swarrow}-5\)
\(-2\) \(4\) \(\stackrel{\nwarrow}{\swarrow}\highlight{2}\)
\(\stackrel{\nwarrow}{\swarrow}-3\)
\(-1\) \(1\) \(\stackrel{\nwarrow}{\swarrow}\highlight{2}\)
\(\stackrel{\nwarrow}{\swarrow}-1\)
\(0\) \(0\) \(\stackrel{\nwarrow}{\swarrow}\highlight{2}\)
\(\stackrel{\nwarrow}{\swarrow}1\)
\(1\) \(1\) \(\stackrel{\nwarrow}{\swarrow}\highlight{2}\)
\(\stackrel{\nwarrow}{\swarrow}3\)
\(2\) \(4\) \(\stackrel{\nwarrow}{\swarrow}\highlight{2}\)
\(\stackrel{\nwarrow}{\swarrow}5\)
\(3\) \(9\)
Table9.2.17

Subsection9.2.4Differentiating Quadratic Functions from Other Functions and Relations

So far, we've seen that the graphs of quadratic functions are parabolas and have a specific, curved shape, which we obtained by making a table of values. We've also seen that they have the algebraic form of \(y=ax^2+bx+c\text{.}\) Here, we'll differentiate quadratic functions from other relations and functions.

Example9.2.18

Determine if each relation represented algebraically is a quadratic function.

  1. \(y+5x^2-4=0\)

  2. \(x^2+y^2=9\)

  3. \(y=-5x+1\)

  4. \(y=(x-6)^2+3\)

  5. \(y=\sqrt{x+1}+5\)

Solution

  1. As \(y+5x^2-4=0\) can be re-written as \(y=-5x^2+4\text{,}\) this equation represents a quadratic function.

  2. The equation \(x^2+y^2=9\) cannot be re-written in the form \(y=ax^2+bx+c\) (due to the \(y^2\) term), so this equation does not represent a quadratic function.

  3. The equation \(y=-5x+1\) represents a linear function, not a quadratic function.

  4. The equation \(y=(x-6)^2+3\) can be re-written as \(y=x^2-12x+39\text{,}\) so this does represent a quadratic function.

  5. The equation \(y=\sqrt{x+1}+5\) does not represent a quadratic function as \(x\) is inside a radical, not squared.
Example9.2.19

Determine if each relation represented graphically could could represent a quadratic function.

  1. <<SVG image is unavailable, or your browser cannot render it>>

    Figure9.2.20

  2. <<SVG image is unavailable, or your browser cannot render it>>

    Figure9.2.21

  3. <<SVG image is unavailable, or your browser cannot render it>>

    Figure9.2.22
Solution

  1. Since this graph has multiple maximum points and minimum points, it is not a parabola and it is not possible that it represents a quadratic function.

  2. This graph looks like a parabola, and it's possible that it represents a quadratic function.

  3. This graph does not appear to be a parabola, but rather looks like a straight line. It's not likely that it represents a quadratic function.
Example9.2.23

Determine if each relation represented in a table could could represent a quadratic function.

  1. \(x\) \(f(x)\)
    \(-3\) \(11\)
    \(-2\) \(7\)
    \(-1\) \(3\)
    \(0\) \(-1\)
    \(1\) \(-5\)
    \(2\) \(-9\)
    \(3\) \(-13\)
    Table9.2.24
  2. \(x\) \(g(x)\)
    \(-3\) \(-9\)
    \(-2\) \(-4\)
    \(-1\) \(-1\)
    \(0\) \(0\)
    \(1\) \(-1\)
    \(2\) \(-4\)
    \(3\) \(-9\)
    Table9.2.25
  3. \(x\) \(p(x)\)
    \(-3\) \(-8\)
    \(-2\) \(4\)
    \(-1\) \(-7\)
    \(0\) \(5\)
    \(1\) \(-6\)
    \(2\) \(6\)
    \(3\) \(-5\)
    Table9.2.26
Solution

  1. The difference betwee each output for consecutive inputs is constant, so it's possible that this function is a linear function. It is therefore not possible that it is a quadratic function.

  2. This pattern looks to follow the pattern of \(-x^2\text{,}\) so it is possible that it is a quadratic function.

  3. This function changes from increasing to decreasing (and visa versa) repeatedly, so it is not possible that it is a quadratic function.

Subsection9.2.5Exercises

Exercises on Graphing Quadratic Functions Using a Table

1

Create a table of ordered pairs and then graph the function \(y=x^2+2\text{.}\)

Solution
\(x\) \(y=x^2+2\)
\(-3\) \((-3)^2+2=11\)
\(-2\) \((-2)^2+2=6\)
\(-1\) \((-1)^2+2=3\)
\(0\) \(0^2+2=2\)
\(1\) \(1^2+2=3\)
\(2\) \(2^2+2=6\)
\(3\) \(3^2+2=11\)

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Table9.2.27
2

Create a table of ordered pairs and then graph the function \(y=x^2+1\text{.}\)

Solution
\(x\) \(y=x^2+1\)
\(-3\) \((-3)^2+1=10\)
\(-2\) \((-2)^2+1=5\)
\(-1\) \((-1)^2+1=2\)
\(0\) \(0^2+1=1\)
\(1\) \(1^2+1=2\)
\(2\) \(2^2+1=5\)
\(3\) \(3^2+1=10\)

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Table9.2.28
3

Create a table of ordered pairs and then graph the function \(y=x^2-5\text{.}\)

Solution
\(x\) \(y=x^2-5\)
\(-3\) \((-3)^2-5=4\)
\(-2\) \((-2)^2-5=-1\)
\(-1\) \((-1)^2-5=-4\)
\(0\) \(0^2-5=-5\)
\(1\) \(1^2-5=-4\)
\(2\) \(2^2-5=-1\)
\(3\) \(3^2-5=4\)

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Table9.2.29
4

Create a table of ordered pairs and then graph the function \(y=x^2-3\text{.}\)

Solution
\(x\) \(y=x^2-3\)
\(-3\) \((-3)^2-3=4\)
\(-2\) \((-2)^2-3=-1\)
\(-1\) \((-1)^2-3=-4\)
\(0\) \(0^2-3=-3\)
\(1\) \(1^2-3=-4\)
\(2\) \(2^2-3=-1\)
\(3\) \(3^2-3=4\)

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Table9.2.30
5

Create a table of ordered pairs and then graph the function \(y=(x-2)^2\text{.}\)

Solution
\(x\) \(y=(x-2)^2\)
\(-3\) \((-3-2)^2=25\)
\(-2\) \((-2-2)^2=16\)
\(-1\) \((-1-2)^2=9\)
\(0\) \((0-2)^2=4\)
\(1\) \((1-2)^2=1\)
\(2\) \((2-2)^2=0\)
\(3\) \((3-2)^2=1\)

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Table9.2.31
6

Create a table of ordered pairs and then graph the function \(y=(x-4)^2\text{.}\) Use \(x\)-values from \(-1\) to \(7\text{.}\)

Solution
\(x\) \(y=(x-4)^2\)
\(-1\) \((-1-4)^2=25\)
\(0\) \((0-4)^2=16\)
\(1\) \((1-4)^2=9\)
\(2\) \((2-4)^2=4\)
\(3\) \((3-4)^2=1\)
\(4\) \((4-4)^2=0\)
\(5\) \((5-4)^2=1\)
\(6\) \((6-4)^2=4\)
\(7\) \((7-4)^2=9\)

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Table9.2.32
7

Create a table of ordered pairs and then graph the function \(y=(x+3)^2\text{.}\) Use \(x\)-values from \(-5\) to \(1\text{.}\)

Solution
\(x\) \(y=(x+3)^2\)
\(-5\) \((-5+3)^2=4\)
\(-4\) \((-4+3)^2=1\)
\(-3\) \((-3+3)^2=0\)
\(-2\) \((-2+3)^2=1\)
\(-1\) \((-1+3)^2=4\)
\(0\) \((0+3)^2=9\)
\(1\) \((1+3)^2=16\)

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Table9.2.33
8

Create a table of ordered pairs and then graph the function \(y=(x+2)^2\text{.}\)

Solution
\(x\) \(y=(x+2)^2\)
\(-4\) \((-4+2)^2=4\)
\(-3\) \((-3+2)^2=1\)
\(-2\) \((-2+2)^2=0\)
\(-1\) \((-1+2)^2=1\)
\(0\) \((0+2)^2=4\)
\(1\) \((1+2)^2=9\)
\(2\) \((2+2)^2=16\)

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Table9.2.34
9

Create a table of ordered pairs and then graph the function \(y=x^2-5x+3\text{.}\)

Solution
\(x\) \(y=x^2-5x-3\)
\(-3\) \((-3)^2-5(-3)+3=27\)
\(-2\) \((-2)^2-5(-2)+3=17\)
\(-1\) \((-1)^2-5(-1)+3=9\)
\(0\) \((0)^2-5(0)+3=3\)
\(1\) \((1)^2-5(1)+3=-1\)
\(2\) \((2)^2-5(2)+3=-3\)
\(3\) \((3)^2-5(3)+3=-3\)
\(4\) \((4)^2-5(4)+3=-1\)
\(5\) \((5)^2-5(5)+3=3\)

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Table9.2.35
10

Create a table of ordered pairs and then graph the function \(y=x^2+7x-1\text{.}\) Use \(x\)-values from \(-8\) through \(1\text{.}\)

Solution
\(x\) \(y=x^2+7x-1\)
\(-8\) \((-8)^2+7(-8)-1=7\)
\(-7\) \((-7)^2+7(-7)-1=-1\)
\(-6\) \((-6)^2+7(-6)-1=-7\)
\(-5\) \((-5)^2+7(-5)-1=-11\)
\(-4\) \((-4)^2+7(-4)-1=-13\)
\(-3\) \((-3)^2+7(-3)-1=-13\)
\(-2\) \((-2)^2+7(-2)-1=-11\)
\(-1\) \((-1)^2+7(-1)-1=-7\)
\(0\) \((0)^2+7(0)-1=-1\)
\(1\) \((1)^2+7(1)-1=7\)

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Table9.2.36
11

Create a table of ordered pairs and then graph the function \(y=-x^2+2x-5\text{.}\)

Solution
\(x\) \(y=-x^2+2x-5\)
\(-3\) \(-(-3)^2+2(-3)-5=-20\)
\(-2\) \(-(-2)^2+2(-2)-5=-13\)
\(-1\) \(-(-1)^2+2(-1)-5=-8\)
\(0\) \(-(0)^2+2(0)-5=-5\)
\(1\) \(-(1)^2+2(1)-5=-4\)
\(2\) \(-(2)^2+2(2)-5=-5\)
\(3\) \(-(3)^2+2(3)-5=-8\)

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Table9.2.37
12

Create a table of ordered pairs and then graph the function \(y=-x^2+4x+2\text{.}\)

Solution
\(x\) \(y=-x^2+4x+2\)
\(-3\) \(-(-3)^2+4(-3)+2=-19\)
\(-2\) \(-(-2)^2+4(-2)+2=-10\)
\(-1\) \(-(-1)^2+4(-1)+2=-3\)
\(0\) \(-(0)^2+4(0)+2=2\)
\(1\) \(-(1)^2+4(1)+2=5\)
\(2\) \(-(2)^2+4(2)+2=6\)
\(3\) \(-(3)^2+4(3)+2=5\)
\(4\) \(-(4)^2+4(4)+2=2\)
\(5\) \(-(5)^2+4(5)^2+2=-3\)

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Table9.2.38
13

Create a table of ordered pairs and then graph the function \(y=4x^2-8x+5\text{.}\)

Solution
\(x\) \(y=4x^2-8x+5\)
\(-3\) \(4(-3)^2-8(-3)+5=65\)
\(-2\) \(4(-2)^2-8(-2)+5=37\)
\(-1\) \(4(-1)^2-8(-1)+5=17\)
\(0\) \(4(0)^2-8(0)+5=5\)
\(1\) \(4(1)^2-8(1)+5=1\)
\(2\) \(4(2)^2-8(2)+5=5\)
\(3\) \(4(3)^2-8(3)+5=17\)

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Table9.2.39
14

Create a table of ordered pairs and then graph the function \(y=2x^2-4x+7\text{.}\)

Solution
\(x\) \(y=2x^2-4x+7\)
\(-3\) \(2(-3)^2-4(-3)+7=37\)
\(-2\) \(2(-2)^2-4(-2)+7=23\)
\(-1\) \(2(-1)^2-4(-1)+7=13\)
\(0\) \(2(0)^2-4(0)+7=7\)
\(1\) \(2(1)^2-4(1)+7=5\)
\(2\) \(2(2)^2-4(2)+7=7\)
\(3\) \(2(3)^2-4(3)+7=13\)

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Table9.2.40