{{ 'ml-label-loading-course' | message }}
{{ toc.name }}
{{ toc.signature }}
{{ tocHeader }} {{ 'ml-btn-view-details' | message }}
{{ tocSubheader }}
{{ 'ml-toc-proceed-mlc' | message }}
{{ 'ml-toc-proceed-tbs' | message }}
Lesson
Exercises
Recommended
Tests
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}. 

{{ article.displayTitle }}

{{ article.intro.summary }}
Show less Show more expand_more
{{ ability.description }} {{ ability.displayTitle }}
Lesson Settings & Tools
{{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }}
{{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }}
{{ 'ml-lesson-time-estimation' | message }}
This lesson will discuss how to graph a particular type of function called a quadratic function.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Maximum and Minimum Points of Graphs

In the coordinate plane, the graphs of three functions and their corresponding equations are shown.
three graphs
Pay close attention to the maximum/minimum point of each graph. Can their coordinates be identified just by looking at the corresponding equation?
Discussion

Parabola and Quadratic Function

All three graphs of the functions presented earlier have a very distinctive form. These curves have a specific name, which now will be properly introduced.

Concept

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 is equidistant from the directrix and the focus
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.

Vertical and horizontal parabolas

Equations of parabolas always contain a variable raised to the second power. This is why the functions that represent vertical parabolas are called quadratic functions.

Concept

Quadratic Function

A quadratic function is a polynomial function of degree that can always be written in the form with Notice that the highest exponent of the independent variable is The graph of any quadratic function is a vertical parabola.
parabola
There are three ways to write a quadratic function.
Name Equation Characteristics
Standard Form and are real numbers, and is the intercept of the parabola.
Vertex Form and are real numbers, and is the vertex of the parabola.
Intercept Form
(also called Factored Form)
and are real numbers, and and are the intercepts of the parabola.
In this lesson, only the vertex and the intercept forms will be discussed.
Discussion

Characteristics of Quadratic Functions

The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.

Concept

Direction of a Parabola

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

direction of a parabola
Concept

Vertex of a Parabola

Because a parabola either opens upward or downward, there is always one point that is the absolute maximum or absolute minimum of the function. This point is called the vertex.
vertex of a parabola showing maximum or minimum button animation
At the vertex, the function changes from increasing to decreasing, or vice versa.
Concept

Axis of Symmetry

An axis of symmetry is the line that divides the graph of a function in two mirrored images. For example, the graph of a quadratic function is a parabola that has an axis of symmetry parallel to the axis and passes through its vertex.
A parabola and its Axis of symmetry
It is also possible for a function to have an axis of symmetry which is not vertical. The following graph shows an example of this.
The graph of the example function has an inclined axis of symmetry

If a function's axis of symmetry is the axis, it is said that it is an even function.

Function 0.8*x^4-3*x^2+1.5 is symmetric with respect to the y-axis.

The concept of an axis of symmetry extends beyond graphs of functions.

Any geometric figure can have an axis of symmetry if there exists a line that divides it into congruent, mirror-image halves.
Concept

Zeros of a Parabola

A parabola can intersect the axis at zero, one, or two points. Since the function's value at an -intercept is always these points are called zeros, or sometimes roots.

zeros of a parabola

intercept

Because all graphs of quadratic functions extend infinitely to the left and right, they each have a -intercept somewhere along the -axis.

y-intercept
Pop Quiz

Identifying Characteristics of a Parabola

Consider the given parabola. Identify its zeros, line of symmetry, or intercept.

identifying characteristics of a parabola
Discussion

Vertex Form of a Quadratic Function

A quadratic function is said to be written in vertex form if it has the following format.

Here, and are real numbers with The value of gives the direction of the parabola. When the parabola faces upward, and when it faces downward. The vertex of the parabola lies at and the axis of symmetry is the vertical line Consider the graph of

Graph of a parabola opening downward
Comparing the generic vertex form with the example function, the values of and can be identified.
These values determine the characteristics of the parabola shown in the graph.
Direction Vertex Axis of Symmetry
and
Since is less than the parabola opens downward. The vertex is located at The axis of symmetry is the vertical line

Extra

Consider other example quadratic functions.
Although these functions do not strictly follow the format for the vertex form, they are said to be written in vertex form because they can easily be rewritten in the desired format.
Function Function Function






Discussion

Graphing a Quadratic Function in Vertex Form

When a quadratic function is written in vertex form, some characteristics of its graph can be identified.

Direction Vertex Axis of Symmetry
It is possible to graph a quadratic function using these characteristics. Consider the function
1
Identify and Plot the Vertex
expand_more
To begin, identify the vertex from the function rule.
Here, and Therefore, the vertex of the parabola is This point can be plotted on a coordinate plane.
The vertex (2,-4) on a coordinate plane
2
Draw the Axis of Symmetry
expand_more

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is here it is

The line x=2 on a coordinate plane
3
Determine and Plot the -intercept
expand_more
Next, determine the intercept. Recall that for all intercepts, the coordinate is Therefore, substitute into the function rule and solve for
Evaluate right-hand side
The intercept of the parabola is This point can be added to the graph.
y-intercept along with the vertex and axis of symmetry on a coordinate plane
4
Reflect the intercept Across the Axis of Symmetry
expand_more
The axis of symmetry divides the parabola into two mirror images. Therefore, points on one side of the parabola can be reflected across the axis of symmetry.
y-intercept is reflected across the axis of symmetry
5
Draw the Parabola
expand_more
Now, with three points plotted, the direction of the parabola can be seen. It appears that the parabola faces upward. Identify the value of in the equation to see if this is correct.
Since which is greater than zero, it can be said that the parabola opens upward. To graph the quadratic function, connect the three points with a smooth curve.
The graph of the parabola (x-2)^2-4 with x-intercepts at (0,0) and (4,0), a vertex at (2,-4), and an axis of symmetry at x=2.
Example

Using the Vertex Form to Draw a Parabola

LaShay loves playing golf.

golf court
She is trying to improve her swing by drawing the parabola that the ball will make. Using her math knowledge, she has calculated the quadratic function that corresponds to this parabola.
Draw the parabola to help LaShay improve her swing!

Answer

parabola

Hint

Identify the vertex of the parabola. Make the graph only for the first quadrant.

Solution

To identify the vertex of the parabola, the given equation can be compared to the general form of a quadratic function written in vertex form.
For the given function, and This means that the vertex of the parabola is Since the parabola illustrates the position of the ball, the first quadrant will only be considered.
vertex

The axis of symmetry of the parabola is the vertical line through the vertex. Therefore, in this case, the equation of the axis of symmetry is

axis of symmetry
The intercept will now be determined and plotted. To do so, will be substituted in the given equation.
Evaluate right-hand side
The parabola intercepts the axis at
y-intercept
Next, the intercept will be reflected across the axis of symmetry.
reflection
Note that for the given function, the value of is Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.
parabola
Example

Writing the Vertex Form of a Quadratic Function Given its Graph

Mark is studying parabolas so that he can help LaShay with her golf swing. He wants to write the vertex form of the quadratic function that corresponds to the given graph.

parabola
Help Mark find the desired equation!

Hint

Start by identifying the vertex of the parabola.

Solution

Recall the format of the vertex form.
In this format, the vertex of the parabola has coordinates Therefore, to state the values of and start by identifying the vertex.
vertex of the parabola is identified
The vertex is the point with coordinates This means that in the equation that corresponds to the given parabola, and
Finally, to find the value of any point on the parabola can be used. For simplicity, the intercept will be used.
y-intercept
The parabola intercepts the axis at Therefore, to find the value of and will be substituted into the equation and the equation will be solved for
Solve for
It has been found that the value of is With this information, the equation of the given parabola can be written in vertex form.
Explore

Zeros of a Parabola

In the coordinate plane below, the graphs of three quadratic functions and their corresponding equations are shown.
Three graphs of different parabolas with their zeros identified
Pay close attention to the zeros of each graph. Can their coordinates be identified just by looking at the corresponding equation?
Discussion

Factored Form of a Quadratic Function

A quadratic function is said to be written in factored form, or intercept form, if it follows a specific format.

Here, and are real numbers with The value of gives the direction of the parabola. When the parabola faces upward, and when it faces downward. The zeros of the parabola are and and the axis of symmetry is the vertical line

Example

Consider the graph of

The graph of parabola y=\dfrac{1}{2}(x-7)(x-13) with the axis of symmetry (x=10) and x-intercepts (x=7 and x=13) shown
Comparing the generic factored form with the example function, the values of and can be identified.
These values determine the characteristics of the parabola shown in the graph.
Direction Zeros Axis of Symmetry
and

Since is greater than the parabola opens upward. The zeros are and Therefore, the parabola intersects the axis at and The axis of symmetry is the vertical line

Extra

Consider other example quadratic functions.
Although these functions do not strictly follow the format of the factored form, they are said to be written in factored form because they can easily be rewritten in the desired format.
Function Function Function






Pop Quiz

Identifying Vertex and Factored Forms

In the following applet, several quadratic functions are expressed in different forms. Are the quadratic functions written in vertex form, factored form, or neither?

The applet randomly generates equations in vertex form, factored form, or neither of these forms
Discussion

Graphing a Quadratic Function in Factored Form

When a quadratic function is written in factored form, some characteristics of its graph can be identified.

Direction Zeros Axis of Symmetry
and

It is possible to graph a quadratic function using these characteristics. Consider the function

1
Identify and Plot the Zeros
expand_more
To begin, identify the zeros and from the function rule.
Here, and Therefore, the intercepts of the parabola are and These points can be plotted on a coordinate plane.
Zeros of parabola on a coordinate plane
2
Determine and Draw the Axis of Symmetry
expand_more
The axis of symmetry is a vertical line with equation By substituting the values of and this equation can be evaluated.
Evaluate right-hand side
The axis of symmetry can now be drawn.
The axis of symmetry on the coordinate plane with the zeros
3
Determine and Plot the Vertex
expand_more
The axis of symmetry is a vertical line that goes through the vertex of the parabola. Since the equation of the axis of symmetry is the coordinate of the vertex is also To find its coordinate, will be substituted into the given equation.
Evaluate right-hand side
The vertex lies at This point can be added to the graph.
The vertex is plotted on the coordinate plane with the zeroes and the axis of symmetry
4
Draw the Parabola
expand_more
Now, with the three points plotted, the direction of the parabola can be seen. It appears that the parabola faces downward. Identify the value of in the equation to see if this is correct.
Since which is less than zero, it can be said that the parabola opens downward. To graph the quadratic function, connect the three points with a smooth curve.
The graph of the parabola -1*(x+1)*(x-5) with x-intercepts at (-1,0) and (5,0), a vertex at (2,9), and an axis of symmetry at x=2.
Example

Using the Factored Form to Draw a Parabola

By now, it is not a secret that LaShay loves playing golf.

golf court
She is once again trying to improve her swing by drawing the parabola that the ball will make. One more time, she uses her math knowledge to calculate the quadratic function that corresponds to this parabola.
Draw the parabola to help LaShay improve her swing!

Answer

parabola

Hint

Identify the zeros of the quadratic function. If necessary, rewrite the function.

Solution

To identify the zeros of the quadratic function, the given equation can be rewritten and compared to the general form of a quadratic function written in factored form.
For the given function, and This means that the zeros of the function are and so the parabola intersects the axis at and
x-intercepts
Next, the parabola's axis of symmetry can be determined.
Evaluate right-hand side
The axis of symmetry is the vertical line with equation
axis of symmetry
Since the axis of symmetry is the vertical line through the parabola's vertex, the coordinate of the vertex is To find its coordinate, can be substituted in the given equation.