Objectives
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Graph quadratic functions and identify their axis of symmetry, and maximum or minimum point.
Section 4.1 Graphing Quadratic Functions (PR1)
Subsection 4.1.1 Activities
Observation 4.1.1.
Quadratic functions have many different applications in the real world. For example, say we want to identify a point at which the maximum profit or minimum cost occurs. Before we can interpret some of these situations, however, we will first need to understand how to read the graphs of quadratic functions to locate these least and greatest values.
Activity 4.1.2.
Use the graph of the quadratic function \(f(x)=3(x-2)^2-4\) to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
Make a table for values of \(f(x) \) corresponding to the given \(x \)-values. What is happening to the \(y\)-values as the \(x\)-values increase? Do you notice any other patterns of the \(y\)-values of the table?
| \(x\) | \(f(x)\) |
|---|---|
| -2 | |
| -1 | |
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
(b)
At which point \((x,y)\) does \(f(x) \) have a minimum value? That is, is there a point on the graph that is lower than all other points?
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The minimum value appears to occur near \((0, 8) \text{.}\)
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The minimum value appears to occur near \(\left(-\dfrac {1}{5}, 10\right) \text{.}\)
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The minimum value appears to occur near \((2, -4) \text{.}\)
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There is no minimum value of this function.
(c)
At which point \((x,y)\) does \(f(x) \) have a maximum value? That is, is there a point on the graph that is higher than all other points?
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The maximum value appears to occur near \((-2, 44) \text{.}\)
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The maximum value appears to occur near \(\left(-\dfrac {1}{5}, 10\right) \text{.}\)
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The maximum value appears to occur near \((2, -4) \text{.}\)
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There is no maximum value of this function.
Definition 4.1.6.
The point at which a quadratic function has a maximum or minimum value is called the vertex. The vertex form of a quadratic function is given by
\begin{equation*}
f(x)=a(x-h)^2+k\text{,}
\end{equation*}
where \((h, k)\) is the vertex of the parabola.
Definition 4.1.7.
The axis of symmetry, also known as the line of symmetry, is the line that makes the shape of an object symmetrical. For a quadratic function, the axis of symmetry always passes through the vertex \((h,k)\) and so is the vertical line \(x = h\text{.}\)
Activity 4.1.8.
Use the given quadratic function, \(f(x)=3(x-2)^2-4\text{,}\) to answer the following:
(a)
(b)
Compare what you got in part \(a\) with the values you found in Activity 4.1.2. What do you notice?
Definition 4.1.9.
The standard form of a quadratic function is given by
\begin{equation*}
f(x)=ax^2+bx+c\text{,}
\end{equation*}
where \(a, b \text{,}\) and \(c \) are real coefficients.
Activity 4.1.10.
Completing the square (see Section 1.5) is a very useful tool or method to convert the quadratic equation in standard form into vertex form. Let’s start with the standard form of \(y=2x^2-4x+7\) and convert it into vertex form.
(a)
Before we begin, isolate the terms with \(x\) on one side of the equation. What equation do you now have?
(b)
Notice that the \(a\)-value is not \(1\text{.}\) Before we complete the square, we will want to factor out the coefficient of the \(x^2\) term. What does your equation look like when factoring out \(2\text{?}\)
(c)
Now apply the "completing the square" steps to determine the constant term that is added to both sides of the equation.
Hint.
Refer back to Definition 1.5.13 to help you determine how to find the constant term. Be careful when the coefficient is not \(1\text{!}\)
(d)
Factor the perfect square trinomial.
(e)
Now isolate \(y\) and then determine the vertex and axis of symmetry.
Remark 4.1.11.
In Activity 4.1.10, we were able to convert from standard form to vertex form. If you were to do this for the general form of \(y=ax^2+bx+c\text{,}\) you will see that the \(x\)-value of the vertex will always be of the form \(-\frac{b}{2a}\text{.}\) For instance, suppose we start with
\begin{equation*}
y=ax^2+bx+c\text{.}
\end{equation*}
Following similar steps as we did in Activity 4.1.10, we would get:
\begin{equation*}
y-c=ax^2+bx
\end{equation*}
\begin{equation*}
y-c=a\left(x^2+\frac{b}{a}x\right)
\end{equation*}
\begin{equation*}
y-c+a\left(\frac{b}{2a}\right)^2=a\left(x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2\right)
\end{equation*}
\begin{equation*}
y-c+a\left(\frac{b}{2a}\right)^2=a\left(x+\frac{b}{2a}\right)^2
\end{equation*}
\begin{equation*}
y=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}
\end{equation*}
In the vertex form, \(y=a(x-h)^2+k\text{,}\) the vertex \((h,k)\) can be identified as:
\begin{equation*}
h=-\frac{b}{2a}
\end{equation*}
\begin{equation*}
k=c-\frac{b^2}{4a}
\end{equation*}
Observation 4.1.12.
Just as with the vertex form of a quadratic, we can use the standard form of a quadratic \(\left(f(x)=ax^2+bx+c\right)\) to find the axis of symmetry and the vertex by using the values of \(a, b \text{,}\) and \(c \text{.}\) Given the standard form of a quadratic, the axis of symmetry is the vertical line \(x=-\dfrac {b}{2a}\) and the vertex is at the point \(\left(-\dfrac{b}{2a},f\left(-\dfrac{b}{2a}\right)\right)\text{.}\)
Activity 4.1.13.
Use the graph of the quadratic function to answer the questions below.
Diagram Exploration Keyboard Controls
(a)
Which of the following quadratic functions could be the graph shown in the figure?
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\(\displaystyle f(x)=x^2+2x+3\)
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\(\displaystyle f(x)=-(x+1)^2+4\)
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\(\displaystyle f(x)=-x^2-2x+3\)
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\(\displaystyle f(x)=(x+1)^2+4\)
(b)
What is the maximum or minimum value?
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\(\displaystyle -1\)
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\(\displaystyle 4\)
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\(\displaystyle -3\)
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\(\displaystyle 1\)
Activity 4.1.15.
Consider the following four graphs of quadratic functions:
Diagram Exploration Keyboard Controls
(a)
(b)
(c)
Which of the graphs above have an axis of symmetry at \(x=2\text{?}\)
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Graph A
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Graph B
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Graph C
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Graph D
(d)
Which of the graphs above represents the function \(f(x)=-(x-2)^2+4\text{?}\)
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Graph A
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Graph B
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Graph C
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Graph D
(e)
Which of the graphs above represents the function \(f(x)=x^2-4x+1\text{?}\)
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Graph A
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Graph B
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Graph C
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Graph D
Remark 4.1.16.
Notice that the maximum or minimum value of the quadratic function is the \(y\)-value of the vertex.
Activity 4.1.17.
A quadratic function \(f(x)\) has a maximum value of \(7\) and axis of symmetry at \(x=-2\text{.}\)
(a)
Sketch a graph of a function that meets the criteria for \(f(x)\text{.}\)
(b)
Was your graph the only possible answer? Try to sketch another graph that meets this criteria.
Remark 4.1.18.
Other points, such as \(x\)- and \(y\)-intercepts, may be helpful in sketching a more accurate graph of a quadratic function.
Activity 4.1.19.
Consider the following two quadratic functions \(f(x)=x^2-4x+20\) and \(g(x)=2x^2-8x+24\) and answer the following questions:
(a)
(b)
(c)
(d)
Now graph both \(f(x)\) and \(g(x)\) and draw a sketch of each graph on one coordinate plane. How are they similar/different?
Remark 4.1.20.
Notice that in Activity 4.1.19, two different functions could have the same vertex and axis of symmetry. When \(|a|\gt0\text{,}\) the graph narrows. When \(0\lt|a|\lt1\text{,}\) the graph widens (refer back to Section 2.4).
Activity 4.1.21.
Answer the following for each quadratic function below.
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Determine if the parabola opens upwards or downwards.
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Determine the coordinates of the vertex of the parabola. Is this point a maximum or a minimum of the quadratic function?
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Determine the axis of symmetry for the parabola.
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Sketch the quadratic function. Be sure to include the vertex along with at least two points on each side of the vertex.
(a)
\(f(x)=x^2+2x-3\)
(b)
\(f(x)=-5(x-3)^2+20\)
(c)
\(f(x)=5x^2+30x+40\)
(d)
\(f(x)=2(x+4)^2-3\)
Subsection 4.1.2 Exercises
Exercises available at
https://tbil.org/preview/precalculus/exercises/#/bank/PR1/
