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Properties and Graphs of Parabolas

Author: Sophia
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what's covered
In this lesson, you will explore aspects of quadratic functions and their graphs. Specifically, this lesson will cover:

Table of Contents

1. Quadratic Functions and Graphs

The general form of a quadratic function is f open parentheses x close parentheses equals a x squared plus b x plus c comma where a comma b, and c are real numbers and a not equal to 0.

The graph of a quadratic function has a shape called a parabola. Some examples are below.

A graph with an x-axis ranging from –1 to 7 and a y-axis ranging from –2 to 6. A U-shaped graph with low point at (3, -1) opens upward, passing through the points (1, 1) to the left and (5, 1) to the right. A graph with an x-axis ranging from –3 to 5 and a y-axis ranging from –2 to 6. A U-shaped graph with high point at (2, 5) opens downward, passing through the points (1, 0) on its left and (3, 0) on its right.

The characteristics of a parabola are shown in the graph below.

A graph with an x-axis and a y-axis ranging from –6 to 6 at intervals of 2. A parabola opens upward from its lowest point (1, -4), which is labeled as 'vertex'. The parabola passes through the points (0, -3), (-1, 0), and (-2, 5) to the left; and (2, -3), (3, 0), and (4, 5) to the right. The point (0, -3) is labeled 'y-intercept' and the points (-1, 0) and (3, 0) are labeled as 'x-intercepts'. A red dashed vertical line with label 'axis of symmetry' passes through the vertex, cutting the graph into two equal pieces.

Note: parabolas can open upward or downward, as you saw earlier.

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two equal parts.

Every parabola has a y-intercept.

Parabolas can have zero, one, or two x-intercepts.

Here is an example problem where we can pull these ideas together.

EXAMPLE

Identify the axis of symmetry, vertex, x-intercepts, and the y-intercept.

A graph with an x-axis ranging from –1 to 7 and a y-axis ranging from –2 to 6. A parabola opens upward from the point (3, -1), passing through the points (1.6, 0) and (4.4, 0).

  • The axis of symmetry is the line x equals 3.
  • The vertex is at the point open parentheses 3 comma space short dash 1 close parentheses.
  • The x-intercepts are approximately open parentheses 1.6 comma space 0 close parentheses and open parentheses 4.4 comma space 0 close parentheses.
  • The y-intercept is approximately open parentheses 0 comma space 3.5 close parentheses.
  • The vertex is a minimum point on the graph.

try it
Consider the quadratic function as shown in the graph:

A graph with an x-axis ranging from –4 to 8 and a y-axis ranging from 0 to 10. A parabola opens upward from the point (3.1, 1), passing through the points (1, 4) and (5.2, 4).
Identify its axis of symmetry, x- and y-intercepts, and its vertex.
The axis of symmetry is x equals 3 comma there is no x-intercept, the y-intercept is open parentheses 0 comma space 8 close parentheses, and the vertex is located at open parentheses 3 comma space 1 close parentheses.

In the next section, we will use the equation to find its vertex.

term to know
Quadratic Function (General Form)
A function of the form f open parentheses x close parentheses equals a x squared plus b x plus c comma where a comma b, and c are real numbers and a not equal to 0.

2. Finding the Vertex of a Quadratic Function

The most basic quadratic function is f open parentheses x close parentheses equals x squared.

Note that its vertex is at open parentheses 0 comma space 0 close parentheses.

A graph with an x-axis ranging from –3 to 3 and a y-axis ranging from 0 to 7. A parabola opens upward from the point (0, 0), passing through the points (-1, 1), (-2, 4) to the left and (1, 1) and (2, 4) to the right.

Now, consider the graph of g open parentheses x close parentheses equals short dash 2 open parentheses x minus 3 close parentheses squared plus 8.

A graph with an x-axis ranging from –2 to 6 and a y-axis ranging from –1 to 8. A U-shaped graph with high point at (3, 8) opens downward, passing through the points (1, 0) on its left and (5, 0) on its right.

From earlier in the course, we know that the graph of g open parentheses x close parentheses is obtained by applying a series of transformations to the graph of f open parentheses x close parentheses.

Namely, f open parentheses x close parentheses is shifted to the right 3 units, reflected over the x-axis, vertically stretched, and shifted up 8 units.

Note that its vertex is located at the point open parentheses 3 comma space 8 close parentheses.

The standard form of a quadratic function is f open parentheses x close parentheses equals a open parentheses x minus h close parentheses squared plus k comma where open parentheses h comma space k close parentheses is the vertex.

try it
Consider the parabola whose equation is f open parentheses x close parentheses equals 3 open parentheses x plus 1 close parentheses squared plus 10.
What are the coordinates of the vertex of the parabola?
open parentheses short dash 1 comma space 10 close parentheses

While the standard form is useful for finding the vertex, how would we find the vertex of a parabola whose equation is in the form f open parentheses x close parentheses equals a x squared plus b x plus c ?

That is, can we find expressions for h and k in terms of a comma b, and c?

Suppose the two forms represent the same parabola. Then, we know they are equal.

a open parentheses x minus h close parentheses squared plus k equals a x squared plus b x plus c Set the two forms equal to each other.
a open parentheses x squared minus 2 h x plus h squared close parentheses plus k equals a x squared plus b x plus c Expand open parentheses x minus h close parentheses squared.
a x squared minus 2 a h x plus a h squared plus k equals a x squared plus b x plus c Distribute.

Since these are two functions of x, the coefficients of each term, x squared comma x, and constant, need to be equal to each other in order for the equation to hold:

Term Left-Hand Side Coefficient Right-Hand Side Coefficient
x squared a a
x short dash 2 a h b
Constant (No x’s) a h squared plus k c

Note that the coefficients of x squared are the same in both forms.

The most important result is the x term: since the coefficients are to be the same, we have short dash 2 a h equals b comma which means that h equals short dash fraction numerator b over denominator 2 a end fraction.

Equating the constant terms, we have a h squared plus k equals c comma which means k equals c minus a h squared. Substituting h equals short dash fraction numerator b over denominator 2 a end fraction comma this becomes k equals c minus a open parentheses short dash fraction numerator b over denominator 2 a end fraction close parentheses squared equals c minus fraction numerator b squared over denominator 4 a end fraction. This is a more complicated formula, and is unnecessary to remember.

This leads us to a formula for the vertex when the quadratic has the form f open parentheses x close parentheses equals a x squared plus b x plus c.

formula to know
Vertex of a Parabola
Given f open parentheses x close parentheses equals a x squared plus b x plus c, where a not equal to 0 comma its vertex is located at the point open parentheses short dash fraction numerator b over denominator 2 a end fraction comma space f open parentheses short dash fraction numerator b over denominator 2 a end fraction close parentheses close parentheses.

EXAMPLE

Find the coordinates of the vertex of the parabola whose equation is f open parentheses x close parentheses equals short dash 2 x squared plus 12 x plus 37.

From the equation, we have a equals short dash 2 comma b equals 12 comma and c equals 37.

The x-coordinate of the vertex is h equals short dash fraction numerator b over denominator 2 a end fraction equals short dash fraction numerator 12 over denominator 2 open parentheses short dash 2 close parentheses end fraction equals 3.

Then, the y-coordinate is f open parentheses 3 close parentheses equals short dash 2 open parentheses 3 close parentheses squared plus 12 open parentheses 3 close parentheses plus 37 equals 55.

Thus, the vertex is located at the point open parentheses 3 comma space 55 close parentheses.

try it
Consider the parabola whose equation is f open parentheses x close parentheses equals short dash 0.2 x squared plus 2 x plus 300.
What are the coordinates of the vertex of the parabola?
The x-coordinate of the vertex is short dash fraction numerator b over denominator 2 a end fraction comma where a equals short dash 0.2 and b equals 2.

Substituting, we have x equals fraction numerator short dash 2 over denominator 2 open parentheses short dash 0.2 close parentheses end fraction equals fraction numerator short dash 2 over denominator short dash 0.4 end fraction equals 5.

Then, the y-coordinate of the vertex is f open parentheses 5 close parentheses equals short dash 0.2 open parentheses 5 close parentheses squared plus 2 open parentheses 5 close parentheses plus 300 equals 305.

Thus, the vertex is located at open parentheses 5 comma space 305 close parentheses.

Putting some aspects of the equation of a parabola together, the vertex can be used to determine the minimum or maximum value of a quadratic function.

term to know
Quadratic Function (Standard [or Vertex] Form)
A quadratic function in the form f open parentheses x close parentheses equals a open parentheses x minus h close parentheses squared plus k comma where open parentheses h comma space k close parentheses is the vertex.

3. Determining the Maximum or Minimum Values of a Quadratic Function

Consider the general form of a quadratic function, f open parentheses x close parentheses equals a x squared plus b x plus c.

In the last section, we saw that the sign of a determines the direction the parabola opens.

If a greater than 0 comma the parabola opens upward. The vertex is the lowest point on the parabola.

A graph with an x-axis and a y-axis. There is an upward-opening parabolic curve with its inverted minimum point slightly above the x-axis.

If a less than 0 comma the parabola opens downward. The vertex is the highest point on the parabola.

A graph with an x-axis and a y-axis. There is a downward-opening parabolic curve with its peak above the x-axis.

This leads to the following generalization.

big idea
Given a quadratic function f open parentheses x close parentheses equals a x squared plus b x plus c colon
  • If a greater than 0 comma the vertex is the lowest point on the graph of f open parentheses x close parentheses. The minimum value of f open parentheses x close parentheses is the y-coordinate of the vertex.
  • If a less than 0 comma the vertex is the highest point on the graph of f open parentheses x close parentheses. The maximum value of f open parentheses x close parentheses is the y-coordinate of the vertex.

EXAMPLE

Consider the equation f open parentheses x close parentheses equals short dash 3 x squared plus 24 x plus 90. Find its vertex and determine whether the minimum or maximum value of f occurs at the vertex.

The x-coordinate of the vertex is h equals fraction numerator short dash b over denominator 2 a end fraction equals fraction numerator short dash 24 over denominator 2 open parentheses short dash 3 close parentheses end fraction equals 4.

The y-coordinate of the vertex is f open parentheses 4 close parentheses equals short dash 3 open parentheses 4 close parentheses squared plus 24 open parentheses 4 close parentheses plus 90 equals 138. Therefore, the vertex is located at the point open parentheses 4 comma space 138 close parentheses.

Since a equals short dash 3 comma which is negative, this means that the maximum value of f occurs at the vertex, and is equal to 138.

Note: this also means that the range of f is open parentheses short dash infinity comma space 138 close square brackets.

watch
To see another example of quadratic functions, check out this video where the aspects of f open parentheses x close parentheses equals 3 x squared minus 12 x minus 15 are discussed.

try it
Consider the function f open parentheses x close parentheses equals 0.5 x squared minus 3 x plus 9.
Does f have a minimum value or a maximum value?
The function f has a minimum value.
What is the minimum or maximum value of f   ?
The x-coordinate of the vertex is short dash fraction numerator b over denominator 2 a end fraction comma where a equals 0.5 and b equals short dash 3.

Substituting, we have x equals short dash fraction numerator open parentheses short dash 3 close parentheses over denominator 2 open parentheses 0.5 close parentheses end fraction equals 3.

Then, the y-coordinate of the vertex is f open parentheses 3 close parentheses equals 0.5 open parentheses 3 close parentheses squared minus 3 open parentheses 3 close parentheses plus 9 equals 4.5.

Thus, the vertex is located at open parentheses 3 comma space 4.5 close parentheses comma and this makes 4.5 the minimum value.

In applications, if a quantity is modeled by a quadratic function, it is possible to determine the minimum or maximum value of that quantity by examining its equation (no need to use a graph).

4. Applications of Minimum or Maximum Values of a Quadratic Function

EXAMPLE

When an object is thrown from the top of a 200-foot tall building, its height after t seconds is modeled by the function h open parentheses t close parentheses equals short dash 16 t squared plus 80 t plus 320 comma where h is measured in feet.

Since h open parentheses t close parentheses is a quadratic function with a less than 0 comma the vertex can be used to determine the maximum height of the object.

The t-coordinate of the vertex is fraction numerator short dash b over denominator 2 a end fraction equals fraction numerator short dash 80 over denominator 2 open parentheses short dash 16 close parentheses end fraction equals 2.5.

Then, the y-coordinate of the vertex is h open parentheses 2.5 close parentheses equals short dash 16 open parentheses 2.5 close parentheses squared plus 80 open parentheses 2.5 close parentheses plus 320 equals 420.

Thus, the vertex is the point open parentheses 2.5 comma space 420 close parentheses.

Thus, the maximum height of the object is 420 feet, and occurs after 2.5 seconds.

try it
An online newspaper has a subscription price of p dollars per month. The business department has determined that the monthly revenue is modeled by the function R open parentheses p close parentheses equals short dash 2 comma 500 p squared plus 160 comma 000 p.
According to this model, is there a minimum or maximum revenue? How do you know?
Since the coefficient of p squared is negative, there is a maximum revenue.
At which price does the minimum or maximum revenue occur?
Use the formula short dash fraction numerator b over denominator 2 a end fraction comma with a equals short dash 2 comma 500 and b equals 160 comma 000.

Then, the price is short dash fraction numerator 160000 over denominator 2 open parentheses short dash 2500 close parentheses end fraction equals 160000 over 5000 equals 32 comma which means $32 in this situation.
What is the minimum or maximum revenue?
Since the leading coefficient of R open parentheses p close parentheses is negative, there will be a maximum revenue at its vertex. The maximum revenue is calculated by R open parentheses 32 close parentheses equals short dash 2500 open parentheses 32 close parentheses squared plus 160000 open parentheses 32 close parentheses equals 2 comma 560 comma 000 comma which means the maximum revenue is $2,560,000.

summary
In this lesson, you learned that the graph of a quadratic function is a parabola. The vertex of the parabola is either the highest or lowest point on the graph, and you are able to find the vertex of a quadratic equation (form f open parentheses x close parentheses equals a x squared plus b x plus c) by using the formula for the vertex of a parabola. If the equation of the quadratic is known, then a formula can be used to determine the coordinates of the vertex, which can then be used to determine the maximum or minimum values of a quadratic function, whose applications include finding the maximum height of an object thrown and evaluating a newspaper's subscription price/revenue model.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know
Quadratic Function (General Form)

A function in the form f open parentheses x close parentheses equals a x squared plus b x plus c where a comma b, and c are real numbers and a not equal to 0.

Quadratic Function (Standard [or Vertex] Form)

A function in the form f open parentheses x close parentheses equals a open parentheses x minus h close parentheses squared plus k comma where open parentheses h comma space k close parentheses is the vertex.

Formulas to Know
Vertex of a Parabola

Given f open parentheses x close parentheses equals a x squared plus b x plus c comma where a not equal to 0 comma its vertex is located at the point open parentheses short dash fraction numerator b over denominator 2 a end fraction comma space f open parentheses short dash fraction numerator b over denominator 2 a end fraction close parentheses close parentheses.

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