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Pre-Algebra View details
5. Quadratic Functions
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eCourses /
Pre-Algebra /
Chapter 8
5. 

Quadratic Functions

Student Learning Objectives:
  • Create and interpret quadratic equations and functions
  • Identify key features of a parabola
  • Create and graph a quadratic function from a table of values

Why
Quadratic functions are useful for modeling various real-world situations, such as determining the optimal angle for launching objects or predicting the path of a car's acceleration. Understanding quadratic expressions helps in analyzing how changes in a function's values impact its graph and solving problems related to design, engineering, and even economics. This lesson highlights how quadratic functions can describe curved relationships, making them applicable in diverse fields such as architecture and optimization problems.

Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
14 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Quadratic Functions
Slide of 14
A type of function that commonly appears in various day-to-day situations is the quadratic function. This lesson will define, examine characteristics, and provide examples of such functions. Enjoy!

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Explore

Analyzing Different Functions

Consider four different graphs of functions. Examine them closely to determine both similarities and differences.

Four graphs of four different functions
Discussion

Introducing Quadratic Expressions

Concept

Quadratic Expression

A quadratic expression is a specific type of polynomial that follows the form ax^2+bx+c. Each term has a specific name.

Quadratic expression ax^2+bx+c where a is the quadratic term, b is the linear term, and c is the constant term

For an expression in this form to be quadratic, a must be a non-zero number. Consider a few examples of quadratic functions.

Quadratic Expression
x^2+3x+5
5x^2+1/4x+23
3/5x^2+1/2x+4.3
Example

Following a Quadratic Path to a New Beginning

Dylan and his crew are traveling to a newly discovered planet named Harmonica where life is possible. The gravity of different space objects cause their space shuttle to follow a curved path.

The Earth and a spaceship on its way to the planet Harmonica
External credits: @catalyststuff

a The path can be represented by the following expression.
3x^2+5x-6x^3+x^2+14+2x^3-11x+4x^3

Combine like terms to determine whether this is a quadratic expression.

b During their flight to Harmonica, Dylan and the crew witnessed a spectacular comet named Celestial Wanderlust. Its path is described by the following expression.
2+7x^3-5x^2-6x+3x^3+6x^2+9+4x

Combine like terms to determine whether this is a quadratic expression.

Hint
a Simplify the expression by combining like terms.
b Simplify the expression by combining like terms.

Solution
a Consider the given algebraic expression.
3x^2+5x-6x^3+x^2+14+2x^3-11x+4x^3

Notice that there are multiple combinations of like terms. Simplify this expression by combining the like terms.

3x^2+ 5x- 6x^3+ x^2+14+ 2x^3- 11x+ 4x^3
CommutativePropAdd

Commutative Property of Addition

3x^2+ x^2+ 5x- 11x- 6x^3+ 2x^3+ 4x^3+14
AddSubTerms

Add and subtract terms

4x^2- 6x- 6x^3+ 6x^3+14
CommutativePropAdd

Commutative Property of Addition

4x^2-6x+ 6x^3- 6x^3+14
DiffZero

6x^3-6x^3=0

4x^2-6x+0+14
IdPropAdd

Identity Property of Addition

4x^2-6x+14

The highest degree of the expression is 2. This means that this is a quadratic expression.

b Consider the given algebraic expression.
2+7x^3-5x^2-6x+3x^3+6x^2+9+4x

Notice that there are multiple combinations of like terms. Simplify this expression by combining the like terms.

2+ 7x^3- 5x^2- 6x+ 3x^3+ 6x^2+ 9+ 4x
CommutativePropAdd

Commutative Property of Addition

2+ 9+ 7x^3+ 3x^3- 5x^2+ 6x^2- 6x+ 4x
AddSubTerms

Add and subtract terms

11+ 10x^3+ x^2- 2x
CommutativePropAdd

Commutative Property of Addition

10x^3+x^2-2x+11

The highest degree of the expression is 3. This means that this is not a quadratic expression.

Pop Quiz

Identifying a Quadratic Expression

Consider the given algebraic expression. Determine whether it is a quadratic expression by combining like terms.

An algebraic expression is randomly generated
Discussion

Parabola

A parabola is a curve that is geometrically defined as the locus of all points equidistant from a line and a point not on the line. The line is called the directrix, and the point is the focus of the parabola. In the following applet, point P is equidistant from the directrix d and the focus F.

parabola

A parabola can be vertical or horizontal. A vertical parabola can open upward or downward. In comparison, a horizontal parabola can open to the left or the right. The graph of a quadratic function is a vertical parabola.

Vertical and horizontal parabolas
Discussion

Direction of a Parabola

A parabola either opens upward or downward. This is the direction of the parabola . If the leading coefficient a of the corresponding equation is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

direction of a parabola
Discussion

Quadratic Function

A quadratic function is a polynomial function of degree 2 that can always be written in the form y=ax^2+bx+c, with a≠ 0. Notice that the highest exponent of the independent variable is 2. The graph of any quadratic function is a vertical parabola.

parabola
Pop Quiz

Identifying a Parabola

Consider the graph of a function. Identify whether the graph shows a parabola or not.

A function graph which may or may not be a parabola
Pop Quiz

Determining the Sign of a Leading Coefficient

Consider the graph of a quadratic function. Is the leading coefficient of the parabola positive or negative?

Parabola with a negative or positive leading coefficient
Example

Building a Bridge

Once Dylan and his crew arrived to Harmonica, they started settling down and building a city. They are planning to build a bridge over the River of Crystallinity that will have shape of a parabola.

A bridge and a table of values
External credits: @‌rawpixel.com
Use the table of values to graph the shape of the bridge. Note that x represents the distance from the base of one side of the bridge, while y represents the height of the bridge over the ground.

Answer
Parabola

Hint
Each pair of numbers in the table of values is an ordered pair that represents the coordinates of a point on the graph. Plot all the points and then connect them with a smooth curve.

Solution
The following table of values is given for the bridge in the shape of a parabola. There, x represents the distance from the base of one side of the bridge, while y represent the height of the bridge over the ground.

x y
0 - 0.6
1 0.5
2 1.4
4 2.6
6 3
8 2.6
10 1.4
12 - 0.6

Note that each pair of values of x and y forms an ordered pair (x,y). These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Points from the table plotted on a coordinate plane

Now, connect the points with a smooth curve. This way the graph of the parabola is drawn.

Parabola graphed
Example

Kangaroo Population

The crew also brought some species of animals from the Earth. They started observing the changes in their populations over time. Here is the table of values that shows the changes in kangaroo population over the period of one year.

Table of values with the pairs x=0, y=30; x=2, y=20; x=4, y=14; x=6, y=12; x=8, y=14; x=10, y=20; x=12, y=30 and kangaroos

Use the table of values to graph the population growth of kangaroos. Note that x represents the number of month since the observation began and y represents the population of kangaroos in the thousands.

Answer
Parabola

Hint
Each pair of numbers in the table of values is an ordered pair that represents the coordinates of a point on the graph. Plot all the points and then connect them with a smooth curve.

Solution
The following table of values represents the population of kangaroos over one year. Here, x represents the number of month since the observation began and y represents the population of kangaroos in thousands.

x y
0 30
2 20
4 14
6 12
8 14
10 20
12 30

Note that each pair of values of x and y forms an ordered pair (x,y). These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Points from the table of values plotted on a coordinate plane

Now, connect the points with a smooth curve. This way the graph of the parabola is obtained.

Parabola graphed
Example

Setting up a Communication System on Harmonica

Another important aspect of settling into life on a new planet is setting up a communication system with Earth. Kevin and his crew began working on installing satellite dishes and a few telescopes. The number of communication devises y on different days x since the work has begun is given by the following table of values.

The planet Harmonica and a table of values with the pairs: x=0, y=0; x=1, y=0.8; x=2, y=3.2' x=5, y=20; x=8, y=51.2; x=10, y=80
Use the table of values to graph the function that represents the number of communication devices on Harmonica. How many communication devices will there be on Harmonica on Day 7? Round the number to the closest tenth.


Hint
Each pair of numbers in the table of values is an ordered pair that represents the coordinates of a point on the graph. Plot all the points and then connect them with a smooth curve.

Solution
Consider the table of values that represents the total number of communication devices y on Harmonica after x days of working on this task.

x y
0 0
1 0.8
2 3.2
3 7.2
4 12.8
5 20
6 28.8

Note that each pair of values of x and y forms an ordered pair (x,y). These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Points from the table of values plotted on a coordinate plane

Now, connect the points with a smooth curve. This way the graph of the parabola is obtained.

Parabola graphed

Finally, use the graph to determine how many communication devices there will be on Harmonica on Day 7. Draw an arrow from x=7 to the parabola and then from the parabola to the y-axis.

Parabola graphed with the arrows

Therefore, there will be about 40 communication devices on Day 7.
Closure

Analyzing Horizontal Parabolas

This lesson focused on vertical parabolas, their characteristics and some examples. With that covered, what about horizontal parabolas? Consider a few examples.

Five different horizontal parabolas
Do horizontal parabolas also represent quadratic functions? That is determined by performing a vertical line test. Recall that a relation is not a function if there are multiple y-values corresponding to the same x-value.
Moving a vertical line across a horizontal parabola

Since the vertical parabola does not pass the vertical line test, it is not a function. This is true for all horizontal parabolas.



Level 1
Level 2
Level 3
Quadratic Functions
Exercises

Consider the following expression. 1-2x+3x^2-4x^3+5-6x+7x^2 Combine like terms to determine whether this is a quadratic expression.

Let's recall that a quadratic expression is an expression with the highest degree equal 2. Consider the given algebraic expression. 1-2x+3x^2-4x^3+5-6x+7x^2 We can see that there are multiple combinations of like terms. Simplify this expression by combining like terms.

The highest degree of the expression is 3. This means that this is not a quadratic expression.

E
Hint & Answer
S
Solution

Consider the following expression. 21x-13x^2+2x^4+7+5x^2-2x^4+8 Combine like terms to determine whether this is a quadratic expression.

Let's recall that a quadratic expression is an expression with the highest degree equal 2. Consider the given algebraic expression. 21x-13x^2+2x^4+7+5x^2-2x^4+8 We can see that there are multiple combinations of like terms. Simplify this expression by combining like terms.

The highest degree of the expression is 2. Therefore, this is a quadratic expression.

E
Hint & Answer
S
Solution

Consider the following graphs of functions.

Four graphs

Which graphs correspond to quadratic functions?

We need to identify which graphs show quadratic functions. Let's recall that the graph of a quadratic function is a parabola.

Definition of a Parabola |-A parabola is a symmetrical curve where each point is equidistant from another point called the locus located between the legs of the parabola.

Now, let's analyze all the given graphs.

We can see that II is a line and not a parabola. Also, III is a horizontal curve, but it is not symmetrical. At the same time, I and IV do meet the definition of a parabola. Therefore, I and IV are the graphs of quadratic functions.

E
Hint & Answer
S
Solution

Consider the following graphs of functions.

Four graphs

Which graphs correspond to quadratic functions?

We need to identify which graphs show quadratic functions. Let's recall that the graph of a quadratic function is a parabola.

Definition of a Parabola |-A parabola is a symmetrical curve where each point is equidistant from another point called the locus located between the legs of the parabola.

Now, let's analyze all the given graphs.

We can see that I is a line and not a parabola. Additionally, IV is a curve whose one leg points down and another leg points up. This means that IV is also not a parabola. Conversely, II and III do meet the definition of a parabola, which means that they are the graphs of quadratic functions.

E
Hint & Answer
S
Solution
Consider a parabola.

Parabola
 Is the leading coefficient of the parabola positive or negative?

What are the coordinates of the vertex?

Let's consider the given parabola.

First, we need to recall the definition of the leading coefficient.

Leading Coefficient |-In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Also, we need to remember two facts.

  • If a leading coefficient is positive, then the parabola points upwards.
  • If a leading coefficient is negative, then the parabola points downwards.

In this case, the parabola points upwards, which means that its leading coefficient is positive.

We also need to determine the coordinates of the vertex of the parabola. Let's consider the parabola and find the coordinates of the lowest point.

Therefore, the coordinates of the vertex are (1,- 4).

E
Hint & Answer
S
Solution
Consider a parabola.

Parabola
 Is the leading coefficient of the parabola positive or negative?

What are the coordinates of the vertex?

Let's consider the given parabola.

First, we need to recall the definition of the leading coefficient.

Leading Coefficient |-In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Also, we need to remember two facts.

  • If a leading coefficient is positive, then the parabola points upwards.
  • If a leading coefficient is negative, then the parabola points downwards.

In this case, the parabola points downwards, which means that its leading coefficient is negative.

We also need to determine the coordinates of the vertex of the parabola. Let's consider the parabola and find the coordinates of the highest point.

Therefore, the coordinates of the vertex are (2,3).

E
Hint & Answer
S
Solution

Quadratic Functions

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