Topics
2. Quadratic Functions and Models
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United States of America
Grades 9-12

2.01 Characteristics of quadratic functions

Lesson
Practice

Introduction

Key features of functions were introduced in lesson  3.04 Characteristics of functions  . We will analyze those characteristics in this lesson and learn about features specific to quadratic functions.

Characteristics of quadratic functions

A quadratic function is a polynomial function of degree 2. A quadratic function can be written in the form f(x)=ax^2+bx+c where a, b, and c are real numbers.

From the graph of a quadratic function, called a parabola, we can identify key features including domain and range, x- and y-intercepts, increasing and decreasing intervals, positive and negative intervals, average rate of change, and end behavior. The parabola also has the following two features that help us identify it, and that we can use when drawing the graph:

Axis of symmetry

A line that divides a figure into two parts, such that the reflection of either part across the line maps precisely onto the other part. For a parabola, the axis of symmetry is a vertical line passing through the vertex.

x
y
Vertex

The point where the parabola crosses the axis of symmetry. The vertex is either a maximum or minimum on the parabola.

x
y

We can determine the key features of a quadratic function from its graph:

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y

This is a graph of the quadratic parent function: f(x)=x^2

  • Axis of symmetry: x=0
  • Vertex: \left(0,0 \right)
  • y-intercept at \left(0, 0\right)
  • x-intercept at \left(0, 0\right)
  • Domain: \left\{x | - \infty \lt x\lt \infty\right\}
  • Range: \left\{y | y\geq 0\right\}
  • As x \to \infty, f(x) \to \infty
  • As x \to - \infty, f(x) \to \infty

We can identify the x-intercepts of some quadratic equations by drawing the graph of the corresponding function.

Three four quadrant coordinate planes. Left coordinate plane titled One x intercept: A parabola with a vertex at (2, 0) and opens upward. Middle coordinate plane titled Two x intercepts: A parabola that passes through points (2, 0) and (negative 4, 0) and opens downward. Right coordinate plane titled No x intercepts: A parabola with a vertex at (0, 1) and opens upward.

The x-intercepts of a quadratic function can also be seen in a table of values, provided the right values of x are chosen and the equation has at least one real x-intercept.

Examples

Example 1

Consider the quadratic function: f(x)=x^2-2x+1

a

Graph the function.

Worked Solution
Create a strategy

We can create a table of values that satisfy f(x) and use it to help graph the function. It can be useful to choose values for x that are positive and negative, as well as x=0:

x-2-101234
f(x)

To complete the table, evaluate the function for each x-value.

Here is how we can obtain f(-2):

f(-2)=(-2)^2-2(-2)+1

f(-2)=4+4+1

f(-2)=9

Repeat this process for each x-value in the table.

x-2-101234
f(x)9410149

Now we can use these points to graph the quadratic function f(x)

Apply the idea
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x
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y
Reflect and check

Having the vertex in your table is useful, since it tells you where the parabola will change direction. Sometimes the table values you select will not include the vertex of the function, depending on the quadratic function being graphed. If you plot your initial table values and find you are unsure where the parabola changes direction, you can add additional values to your table until you can identify where f(x) changes direction.

Note that the quadratic function has one x-intercept, at x=1.

b

State the axis of symmetry.

Worked Solution
Create a strategy

The axis of symmetry passes through the point where the y-values change from decreasing to increasing.

Apply the idea
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f(x)

The axis of symmetry is x=1.

Example 2

Consider the graph of the quadratic function g(x):

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g(x)
a

Find the x-intercepts and y-intercept.

Worked Solution
Create a strategy

To find the x-intercepts, locate the places where the parabola crosses the x-axis.

To find the y-intercept, locate the place where the parabola crosses the y-axis.

Apply the idea

We can identify the intercepts on the graph:

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y

From the graph we can see the there are two x-intercepts at \left(-4, 0\right) and \left(2, 0 \right), and there is one y-intercept at (0,8).

b

Determine the domain and range.

Worked Solution
Create a strategy

To find the domain of g(x), we want to find all possible x-values for which g(x) could be graphed.

To find the range, we want to find all possible values of g(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of g(x).

Apply the idea

We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues infinitely in either x direction.

This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues infinitely in the negative y direction.

Domain: - \infty < x < \infty

Range: - \infty < y \leq 9

c

Identify each interval where the function is increasing or decreasing.

Worked Solution
Create a strategy

g(x) is increasing where the output values become larger as input values become larger.

g(x) is decreasing where the output values become smaller as its input values become larger.

Parabolas change from increasing to decreasing or decreasing to increasing about the x-value of the vertex.

Apply the idea

We can see from the graph of g(x) that the parabola is increasing up until x=-1 and decreasing after x=-1.

The function is neither increasing or decreasing at the vertex.

Increasing: x < -1

Decreasing: x > -1

d

Identify each interval where the function is either positive or negative.

Worked Solution
Create a strategy

In order to find the intervals for where the function is positive, we want to find the set of x-values where the parabola lies above the x-axis.

Similarly, to find the intervals where the function is negative, we want to find the set of x-values where the parabola lies below the x-axis.

The x-intercepts are where the function crosses the x-axis, where the function is equal to 0. These values will help us determine where the function is positive and negative.

Apply the idea
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y

We can see that the parabola lies above the x-axis between x=-4 and x=2. We can also see that the parabola lies below the x to the left of x=-4 and to the right of x=2.

Positive: -4 < x < 2

Negative: x < -4, x > 2

e

State the end behavior of the function.

Worked Solution
Create a strategy

In order to find the end behavior, we want to find the y-value that the function approaches as the x-values approach positive and negative infinity. We can see on the graph that as the x-values on the parabola get larger in either direction, the parabola continues downwards or towards -\infty.

Apply the idea

x \to \infty, y \to - \infty

x \to - \infty, y \to - \infty

Example 3

The graph shows the height, y (in feet), of a softball above ground x seconds after it was thrown in the air.

Softball throw
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\text{Time in seconds, }x
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\text{Height in feet, }y
a

Find the y-intercept and describe what it means in context.

Worked Solution
Create a strategy

We want to find the place where the parabola crosses the y-axis.

Once we find the y-intercept, we want to connect this to the context of the softball. Since the y-axis represents the height of the softball in feet above the ground, we can use it to identify the height of the softball at 0 seconds.

Apply the idea

We can identify the y-intercept on the graph:

Softball throw
0.5
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\text{Time in seconds, }x
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\text{Height in feet, }y

The y-intercept is \left(0, 6\right).

The y-intercept tells us that the softball was thrown from a height of 6 feet above the ground.

b

Find the value of the x-intercept and describe what it means in context.

Worked Solution
Create a strategy

We want to find the place where the parabola crosses the x-axis.

Once we find the x-intercept, we want to connect this to the context of the softball. Since the x-axis represents the time in seconds after being thrown, we can use it to identify how many seconds the softball hits the ground.

Apply the idea

We can identify the x-intercept on the graph:

Softball throw
0.5
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3.5
\text{Time in seconds, }x
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\text{Height in feet, }y

The x-intercept is \left(3, 0\right).

The x-intercept tells us that the softball hits the ground 3 seconds after it was thrown in the air.

c

Find the value of the vertex and describe what it means in context.

Worked Solution
Create a strategy

In order to find the vertex, we want to find the maximum point of the parabola.

Once we find the vertex, we want to connect this to the context of the softball. We know that the x-value of the vertex represents time in seconds after the softball is thrown and the y-value of the vertex represents the height of the softball above ground in feet.

Apply the idea

We can identify the vertex on the graph:

Softball throw
0.5
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\text{Time in seconds, }x
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\text{Height in feet, }y

The vertex is \left(1.25, 12 \right).

After 1.25 seconds, the softball reaches a maximum height of 12 feet above the ground.

d

State the domain and describe what it means in context.

Worked Solution
Create a strategy

The domain of the context should be reasonable. We can use the graph of the function to determine the domain and explain its meaning in context.

Apply the idea

Domain: 0 \leq x \leq 3

The domain of the function starts at x=0 seconds when the softball was recorded from where it was initially thrown. The domain of the function ends at x=3 seconds when the softball lands on the ground.

e

Determine the average rate of change of the function over the interval 2.5 \leq x \leq 3 and describe what it means in context.

Worked Solution
Create a strategy

To find the average rate of change, we can use the endpoints of the interval with the average rate of change formula:

\dfrac{f(b)-f(a)}{b-a}The endpoints of the interval are at f(2.5) and f(3).

Apply the idea

We can use the graph to find that f(2.5)=6 and f(3)=0. Inputting these points into the average rate of change formula gives us:

\displaystyle m\displaystyle =\displaystyle \dfrac{f(b)-f(a)}{b-a}Average rate of change formula
\displaystyle m\displaystyle =\displaystyle \dfrac{f(3)-f(2.5)}{3-2.5}Substitute b=3 and a=2.5
\displaystyle m\displaystyle =\displaystyle \dfrac{0-6}{3-2.5}Substitute f\left(3\right)=0 and f\left(2.5\right)=6
\displaystyle m\displaystyle =\displaystyle -12Evaluate the subtraction and division

Therefore, the average rate of change of the ball is -12 feet per second from 2.5 to 3 seconds. This means that the ball's vertical distance was decreasing at this time.

Idea summary

From the graph of a quadratic function, we can identify key features including:

  • Domain and range
  • x- and y-intercepts
  • Increasing and decreasing intervals
  • Positive and negative intervals
  • Average rate of change over various intervals
  • End behavior
  • Vertex
  • Axis of symmetry

Outcomes

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

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