# 10.1: Quadratic Functions and Their Graphs

**Basic**Created by: CK-12

**Practice**Quadratic Functions and Their Graphs

Suppose

### Graphs of Quadratic Functions

Previous Concepts introduced the concept of factoring quadratic trinomials of the form **standard form for a quadratic equation.** The most basic quadratic equation is *quadrare,* meaning “to square.” By creating a table of values and graphing the ordered pairs, you find that a quadratic equation makes a **parabola.**

#### Example A

Graph the most basic quadratic equation,

–2 | 4 |

–1 | 1 |

0 | 0 |

1 | 1 |

2 | 4 |

**Solution:**

By graphing the points in the table, you can see that the shape is approximately like the graph below. This shape is called a **parabola.**

**The Anatomy of a Parabola**

A parabola can be divided in half by a vertical line. Because of this, parabolas have **symmetry.** The vertical line dividing the parabola into two equal portions is called the **line of symmetry.** All parabolas have a **vertex,** the ordered pair that represents the bottom (or the top) of the curve.

The vertex of a parabola is the ordered pair

Because the line of symmetry is a vertical line, its equation has the form *the* *coordinate of the vertex.*

As with linear equations, the

An equation of the form

If **upward.** The vertex will be a **minimum.**

If **downward.** The vertex will be a **maximum.**

The variable **leading coefficient** of the quadratic equation. Not only will it tell you if the parabola opens up or down, but it will also tell you the width.

If **narrow** about the line of symmetry.

If **wide** about the line of symmetry.

#### Example B

*Find the x-intercepts of the quadratic function y=x2+5x−6.*

**Solution:**

To find the

This means that

Thus the

**Finding the Vertex of a Quadratic Equation in Standard Form**

The

#### Example C

*Determine the direction, shape and vertex of the parabola formed by*

**Solution:**

The value of

- Because
a is negative, the parabola opens downward. - Because
a is between –1 and 1, the parabola is wide about its line of symmetry. - Because there is no
b term,b=0 . Substituting this into the equation for thex -coordinate of the vertex,x=−b2a=−02a=0 . (Note: It does not matter whata equals; sinceb=0 , the fraction equals zero.) To find they -coordinate, substitute thex -coordinate into the equation:

The vertex is

**Domain and Range**

Several times throughout this textbook, you have experienced the terms **domain** and **range.** Remember:

- Domain is the set of all inputs (
x− coordinates). - Range is the set of all outputs (
y− coordinates).

The domain of every quadratic equation is all real numbers

If

If *coordinate of the vertex.*

#### Example D

*Find the range of the quadratic function y=−2x2+16x+5.*

**Solution:**

To find the range, we must find the

Since the

Now, substitute 4 into the function:

The

Thus, the range is

### Video Review

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### Guided Practice

*Determine the direction, vertex and range of y=7x2+14x−9.*

**Solution:**

Since

Now, substitute

Thus, the vertex is (-1, -16).

Since the parabola faces up, and the

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Graphs of Quadratic Functions (16:05)

- Define the following terms in your own words.
- Vertex
- Line of symmetry
- Parabola
- Minimum
- Maximum

- Without graphing, how can you tell if
y=ax2+bx+c opens up or down?

Graph the following equations by making a table. Let

y=2x2 y=−x2 y=x2−2x+3 y=2x2+4x+1 y=−x2+3 y=x2−8x+3 y=x2−4

Does the graph of the parabola open up or down?

y=−2x2−2x−3 y=3x2 y=16−4x2

Find the

x2−14x+45=0 8x2−16x−42=0 4x2+16x+12=0 x2+2x−15=0

Graph the following functions by making a table of values. Use the vertex and

y=4x2−4 y=−x2+x+12 y=2x2+10x+8 y=12x2−2x y=x−2x2 y=4x2−8x+4

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.1.

-intercept of a parabola

As with linear equations, the**of a parabola are where the graph intersects the -axis. The -value is zero at the**

*-intercepts***.**

*-intercepts*leading coefficient of a parabola

The variable in the equation is called the**of the quadratic equation.**

*leading coefficient*minimums and maximums of a parabola

An equation of the form forms a parabola. If is positive, the parabola will open**The vertex will be a**

*upward.***If is negative, the parabola will open**

*minimum.***The vertex will be a**

*downward.*

*maximum.*symmetry of a parabola

A parabola can be divided in half by a vertical line. Because of this, parabolas have**. The vertical line dividing the parabola into two equal portions is called the line of**

*symmetry***.**

*symmetry*vertex of a parabola

All parabolas have a**, the ordered pair that represents the bottom (or the top) of the curve. The line of symmetry always goes through the vertex. The**

*vertex***is the ordered pair .**

*vertex of a parabola*Coefficient

A coefficient is the number in front of a variable.Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.domain

The domain of a function is the set of -values for which the function is defined.Horizontal shift

A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right.Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".quadratic function

A quadratic function is a function that can be written in the form , where , , and are real constants and .standard form

The standard form of a quadratic function is .Symmetry

A figure has symmetry if it can be transformed and still look the same.Vertex

The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.vertical axis

The vertical axis is also referred to as the -axis of a coordinate graph. By convention, we graph the output variable on the -axis.### Image Attributions

## Description

## Learning Objectives

Here you'll learn about the characteristics of a parabola, including its range and domain.

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## Date Created:

Feb 24, 2012## Last Modified:

Oct 01, 2015## Vocabulary

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