Skip to main content
Logo image

Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

TBIL Community

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.
🔗
Quadratic function that opens upward
Figure 4.1.3.
🔗
(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?
🔗
Table 4.1.4.
🔗
\(x\) \(f(x)\)
-2
-1
0
1
2
3
4
5
Answer.
Table 4.1.5.
🔗
\(x\) \(f(x)\)
-2 44
-1 23
0 8
1 -1
2 -4
3 -1
4 8
5 23
🔗
🔗
🔗
(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?
  1. The minimum value appears to occur near \((0, 8) \text{.}\)
    🔗
    🔗
  2. The minimum value appears to occur near \(\left(-\dfrac {1}{5}, 10\right) \text{.}\)
    🔗
    🔗
  3. The minimum value appears to occur near \((2, -4) \text{.}\)
    🔗
    🔗
  4. There is no minimum value of this function.
    🔗
    🔗
🔗
Answer.
🔗
🔗
(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?
  1. The maximum value appears to occur near \((-2, 44) \text{.}\)
    🔗
    🔗
  2. The maximum value appears to occur near \(\left(-\dfrac {1}{5}, 10\right) \text{.}\)
    🔗
    🔗
  3. The maximum value appears to occur near \((2, -4) \text{.}\)
    🔗
    🔗
  4. There is no maximum value of this function.
    🔗
    🔗
🔗
Answer.
🔗
🔗
🔗

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)
Applying Definition 4.1.6, what is the vertex and axis of symmetry of \(f(x)\text{?}\)
🔗
  1. vertex: \((2,-4)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
  2. vertex: \((-2,4)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  3. vertex: \((-2,-4)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  4. vertex: \((2,4)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
Answer.
🔗
🔗
(b)
Compare what you got in part \(a\) with the values you found in Activity 4.1.2. What do you notice?
🔗
Answer.
Students will probably notice that the vertex is also the minimum of the quadratic function and that the axis of symmetry goes through the minimum.
🔗
🔗
🔗
🔗

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?
🔗
Answer.
\(y-7=2x^2-4x\)
🔗
🔗
🔗
(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 \(2\text{?}\)
🔗
Answer.
\(y-7=2(x^2-2x)\)
🔗
🔗
🔗
(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 \(1\text{!}\)
🔗
🔗
Answer.
By using the "completing the square" steps, students should get:
\begin{equation*} y-7+(2)=2(x^2-2x+(1)) \end{equation*}
\begin{equation*} y-5=2(x^2-2x+1) \end{equation*}
🔗
🔗
🔗
(d)
Factor the perfect square trinomial.
🔗
Answer.
\begin{equation*} y-5=2(x-1)^2 \end{equation*}
🔗
🔗
🔗
(e)
Now isolate \(y\) and then determine the vertex and axis of symmetry.
🔗
Answer.
\(y=2(x-1)^2+5\)
🔗
Axis of Symmetry: \(x=1\)
🔗
Vertex: \((1,5)\)
🔗
🔗
🔗
🔗

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.
🔗
Quadratic function that opens downward with vertex (-1,4)
Figure 4.1.14.
🔗
(a)
Which of the following quadratic functions could be the graph shown in the figure?
  1. \(\displaystyle f(x)=x^2+2x+3\)
    🔗
    🔗
  2. \(\displaystyle f(x)=-(x+1)^2+4\)
    🔗
    🔗
  3. \(\displaystyle f(x)=-x^2-2x+3\)
    🔗
    🔗
  4. \(\displaystyle f(x)=(x+1)^2+4\)
    🔗
    🔗
🔗
Answer.
B or C because they both have a negative in the front, which would indicate that the graph is pointing downward.
🔗
🔗
🔗
(b)
What is the maximum or minimum value?
  1. \(\displaystyle -1\)
    🔗
    🔗
  2. \(\displaystyle 4\)
    🔗
    🔗
  3. \(\displaystyle -3\)
    🔗
    🔗
  4. \(\displaystyle 1\)
    🔗
    🔗
🔗
Answer.
B. Instructors might need to distinguish what a max or min value is versus where the max or min value is located.
🔗
🔗
🔗
🔗

Activity 4.1.15.

Consider the following four graphs of quadratic functions:
🔗
🔗

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 \(f(x)\text{.}\)
🔗
Answer.
Note that this is just one possible sketch students might draw.
A sketch of one possible quadratic that fits the criteria
Figure 4.1.18.
🔗
🔗
🔗
🔗
(b)
Was your graph the only possible answer? Try to sketch another graph that meets this criteria.
🔗
Answer.
No. There are many possible graphs students could sketch, but each would have a maximum point at \((-2,7)\text{.}\) The graph could be transformed by horizontal stretches or compressions.
🔗
🔗
🔗
🔗

Remark 4.1.19.

Other points, such as \(x\)- and \(y\)-intercepts, may be helpful in sketching a more accurate graph of a quadratic function.
🔗
🔗

Activity 4.1.20.

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)
Applying Definition 4.1.9, what is the vertex and axis of symmetry of \(f(x)\text{?}\)
🔗
  1. vertex: \((2,-16)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
  2. vertex: \((-2,16)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  3. vertex: \((-2,-16)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  4. vertex: \((2,16)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
Answer.
🔗
🔗
(b)
Applying Definition 4.1.9, what is the vertex and axis of symmetry of \(g(x)\text{?}\)
🔗
  1. vertex: \((2,-16)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
  2. vertex: \((-2,16)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  3. vertex: \((-2,-16)\text{;}\) axis of symmetry: \(x=-2\)
    🔗
    🔗
  4. vertex: \((2,16)\text{;}\) axis of symmetry: \(x=2\)
    🔗
    🔗
Answer.
🔗
🔗
(c)
What do you notice about \(f(x)\) and \(g(x)\text{?}\)
🔗
Answer.
They have the same vertex and axis of symmetry.
🔗
🔗
🔗
(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?
🔗
Answer.
By graphing the two functions, students should be able to see that although \(f(x)\) and \(g(x)\) have the same vertex and axis of symmetry, they are different functions/graphs. The graph of \(g(x)\) is "skinnier."
Figure 4.1.21.
🔗
🔗
🔗
🔗
🔗

Remark 4.1.22.

Notice that in Activity 4.1.20, 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 4.1.23.

Answer the following for each quadratic function below.
  1. Determine if the parabola opens upwards or downwards.
    🔗
    🔗
  2. Determine the coordinates of the vertex of the parabola. Is this point a maximum or a minimum of the quadratic function?
    🔗
    🔗
  3. Determine the axis of symmetry for the parabola.
    🔗
    🔗
  4. Sketch the quadratic function. Be sure to include the vertex along with at least two points on each side of the vertex.
    🔗
    🔗
🔗
🔗
🔗

Subsection 4.1.2 Exercises

🔗
🔗
PrevTopNext