CH 9. Quadratic Equations and Functions - Mr park's...

CH 9. Quadratic Equations and Functions Algebra I

9.1: Graph 9.2: Graph 9.3: Solve Quadratic Equations by Graphing

9.4: Use Square Roots to Solve Quadratic Equations

9.5: Solve Quadratic Equations by Completing the Square

9.6: Solve Quadratic Equations by Quadratic Formula

9.7: Interpret the Discriminant

9.8: Compare Linear, Exponential, and Quadratic Models Prerequisite Skills

1. The x-coordinate of a point where a graph crosses the x-axis is a(n) ________.

2. A(n) __________ is a function of the form where , and .

3. Evaluate the expression.

a. √

b. √

c. √

KEY VOCABULARY

Quadratic function Vertex Quadratic equation

Parabola Axis of symmetry Completing the square

Parent quadratic function Minimum value Quadratic formula

Maximum value Discriminant


9.1 Graph

A quadratic function is a nonlinear function that can be written in the standard form

where . Every quadratic function has a U-shaped graph called a

parabola. In this lesson, you will graph quadratic function where .

Parent Quadratic Function

The most basic quadratic function in the family of quadratic functions, called the

parent quadratic function, is . The graph of is shown below.

The lowest or

highest point on a

parabola is the

vertex. The vertex

of the graph of

is (0, 0)

The line that passes through the

vertex and divides the parabola into

two symmetric parts is called the

axis of symmetry. The axis of

symmetry for the graph of

is the y-axis, .

Graph | |

STEP 1 Make a table of values for .

x 0 1 2

y 12 3 0 3 12

STEP 2 Plot the points from the table.

STEP 3 Draw the smooth curve through the points

STEP 4 Compare the graph of and .

Both graphs open up and have the same vertex,

(0, 0), and axis of symmetry, . The graph

of is narrower than the graph of

because the graph of is vertical stretch

(by factor of 3) of the graph of .

Graph | |

STEP 1 Make a table of values for

.

x 0 2 4

y 0


STEP 3 Draw the smooth curve through the points.

STEP 4 Compare the graph of

and . Both graphs have the same

vertex, (0, 0), and axis of symmetry, . However, the graph of

is wider than the graph of and it opens down. This is because the graph

of

is vertical shrink (by factor of

) with a reflection in the x-axis of

the graph of .


GRAPHING QUADRATIC FUNCTIONS

Example 1 and 2 suggest the following general result: a parabola opens up when the

coefficient of is positive and opens down when the coefficient of is negative.

Graph

STEP 1 Make a table of values for .

x 0 1 2

y 9 6 5 6 9



STEP 4 Compare the graph of and .

Both graphs open up and have the same axis of

symmetry, . However, the vertex of the

graph of , (0, 5) is different than the

graph of , (0, 0), because the graph of

is vertical translation (of 5 units up)

of the graph of .

Ex) Graph the function. Compare the graph with the graph of .


Graph

STEP 1 Make a table of values for

.

x 0 2 4

y 4 4



STEP 4 Compare the graph of

and . Both graphs open up and

have the same axis of symmetry, . However, the graph of

is

wider and has a lower vertex than the graph of , because the graph of

is vertical shrink and a vertical translation of the graph of .

Ex) Graph the function. Compare the graph with the graph of .


Ex) A solar though has a reflective parabolic surface that is used to collect solar energy. The

sun’s rays are reflected from the surface toward a pipe that carries water. The heated water

produces steam that is used to produce electricity.

The graph of the function

models the

cross section of the

reflective surface where x

and y are measured in

meter. Use the graph to

find the domain and range

of the function in this

situation.

Ex) A cross section of the parabolic surface of the antenna shown can be modeled by the

graph of the function where x and y are measured in meters.

a. Find the domain and range of the function in this situation.


9.2 Graph

You can use the properties below to graph any quadratic function.

Properties of the Graph of Quadratic Function

The graph of is a parabola that:

Opens up if and opens down if .

Is narrower than the graph of if | |

and wider if | | .

Has an axis of symmetry of

.

Has a vertex with an x-coordinate of

.

Has a y-intercept of c. so, the point (0, c) is on

the parabola.

Ex) Find the axis of symmetry and the vertex, from the given function.

Graph

STEP 1 Determine whether the parabola opens up or down. Because a > 0, the parabola

opens up.

STEP 2 Find and draw the axis of symmetry:

( ) .

STEP 3 Find and plot the vertex.

The x-coordinate, substitute 1 for x in the

function and simplify.

( ) ( )

So, the vertex is ( )

STEP 4 Plot two points. Choose two x-values less than

the x-coordinate of the vertex.

Then find the corresponding y-values.

x 0

y 2 11

STEP 5 Reflect the points plotted in Step 4, in the axis of symmetry.

STEP 6 Draw a parabola through the plotted points.


Minimum and Maximum Values

For , the y-coordinate of the vertex is the minimum value of the

function if or the maximum value of the function if .

Ex) Tell whether the given function has a minimum value or a maximum value. Then find the

minimum or maximum value.

( )

( )


9.3 Solve Quadratic Equations By Graphing

A quadratic equation is an equation that can be written in the standard form

where .

In Chapter 8, you used factoring to solve a quadratic equation. You can also use graphing

to solve a quadratic equation. Notice that the solutions of the equation

are the x-intercepts of the graph of the related function .

Solve by Factoring

( )( )

Solve by Graphing

To solve ,

graph . From

the graph you can see that the

x-intercepts are 1 and 5.

To solve a quadratic equation by graphing, first write the equation in standard form,

. Then graph the related function . The x-intercepts

of the graph are the solutions, or roots, of .

Number of Solutions of a Quadratic Equation

A quadratic equation has

two solutions if the graph

of its related function has

two x-intercepts.

A quadratic equation has

one solution if the graph of

its related function has one

x-intercept.

A quadratic equation has no

real solution if the graph of

its related function has no x-

intercepts.

Ex) Find the solutions of given equation (graph).


9.4 Use Square Roots to Solve Quadratic Equations

To use square roots to solve a quadratic equation of the form , first isolate

on one side to obtain . Then use the following information about the solutions of

to solve the equation.

Solve by Taking Square Roots

If d > 0, then has two solutions:

√

If d = 0, then has one solution:

If d < 0, then has no solution.

Ex) Solve quadratic equations

a.

b.

c.

d.

e.

f.

g.

SIMPLIFYING SQUARE ROOTS

In cases where you need to take the square root of a fraction whose numerator and

denominator are perfect squares, the radical can be written as a fraction. For example, √

can be written as

because (

)

.

Ex) Solve a quadratic equation.

a. ( )

b. ( )

c. ( )


9.5 Solve Quadratic Equations by Completing the Square

For an expression of the form , you can add a constant c to the expression so that

the expression is a perfect square trinomial.

This process is called completing the square

COMPLETING THE SQUARE

Words To complete the square for the expression , add the square of half the

coefficient of the term bx.

Algebra (

)

= (

)

Ex) Find the value of c that makes the expression a perfect square trinomial. Then write the

expression as the square of a binomial.

1)

2)

3)

4)

SOLVING EQUATIONS

The method of completing the square can be used to solve any quadratic equation. To use

completing the square to solve a quadratic equation, you must write the equation in the

form .

Ex) Solve the equation by completing the square.

1)


2)

3)

4)

5)

Word Problem

You decide to use chalkboard paint to create a chalkboard

on a door. You want the chalkboard to have a uniform

border as shown. You have enough chalkboard paint to

cover 6 square feet. Find the width of the border to the

nearest inch.


9.6 Solve Quadratic Equations by the Quadratic Formula

By completing the square for the quadratic equation , you can develop a

formula that gives the solutions of any quadratic equation in standard form. This formula

is called the quadratic formula.

The Quadratic Formula

The solutions of the quadratic equation are

√

where and .

Ex) Solve the equation.

1)

2)

3)

4)

5)


CONCEPT SUMMARY

Methods for Solving Quadratic Equations

Method Lesson(s) When to Use

Factoring 9.4~9.8 Use when a quadratic equation can be factored easily.

Graphing 10.3 Use when approximate solutions are adequate.

Finding square

roots 10.4

Use when solving an equation that can be written in the

form .

Completing the

square 10.5

Can be used for any quadratic equation

but is simplest to apply when and b is an even

number.

Quadratic formula 10.6 Can be used for any quadratic equation.

Ex) Tell what method you would use to solve the quadratic equation. Explain your choice(s).

1)

2)

3)

4)

5)

6)


In the quadratic formula, the expression is called the discriminant of the

associated equation .

√

Discriminant

Because the discriminant is under the radical symbol, the value of the discriminant can be

used to determine the number of solutions of a quadratic equation and the number of x-

intercepts of the graph of the related function.

KEY CONCEPT

Using the Discriminant of

Value of the

discriminant

Number of

solutions Two solutions One solution No solution

Graph of

Two x-intercepts One x-intercept No x-intercept

Ex) Tell whether the equation has two solutions, one solution, or no solution.

1)

2)

3)


4)

5)

6)

7) –

Ex) Find the number of x-intercepts of the graph of the function.

1)

2)

3)

4)


9.7 Solve Systems with Quadratic Equations

You have solved systems of linear equations using the graph-and-check method and using

the substitution method. You can use both of these techniques to solve a system of equations

involving nonlinear equations, such as quadratic equations.

Recall that the substitution method consists of the following three steps.

STEP 1 Solve one of the equations for one of its variables.

STEP 2 Substitute the expression from Step 1 into the other equation and solve for the

other variable.

STEP 3 Substitute the value from Step 2 into one of the original equations and solve.

POINT OF INTERSECTION

When you graph a system of equations, the graphs intersect at each solution of the system.

For a system consisting of a linear equation and a quadratic equation the number of

intersections, and therefore solutions, can be zero, one, or two.

KEY CONCEPT

Systems With One Linear Equation and One Quadratic Equation

There are three possibilities for the number of points of intersection

No Solution One Solution Two Solution


Ex) Use the substitution method to solve the system

1)

2)

3)

4)


9.8 Compare Linear, Exponential, and Quadratic Models

So far you have studied linear functions, exponential functions, and quadratic functions.

You can use these functions to model data.

KEY CONCEPT

Linear, Exponential, and Quadratic Functions

Linear Function Exponential Function Quadratic Function

Ex) Use a graph to tell whether the ordered pairs represent a linear function, an exponential

function, or a quadratic function.

(

) (

) (

) ( ) ( )

( ) ( ) ( ) ( ) ( )


( ) ( ) ( ) ( ) ( )

DIFFERENCES AND RATIOS

A table of values represents a linear function if the differences of successive y-values are all

equal. A table of values represents an exponential function if the ratios of successive y-values

are all equal. In both cases, the increments between successive x-values need to be equal.

Use differences or ratios to tell whether the table of values represents a linear function, an

exponential function, or a quadratic function. Extend the table to find the y-value for the next

x-value.

1) x −2 −1 0 1 2

y −6 −6 −4 0 6


2) x −2 −1 0 1 2

y −2 1 4 7 10

WRITING AN EQUATION

When you decide that a set of ordered pairs represents a linear, an exponential, or a quadratic

function, you can write an equation for the function. In this lesson, when you write an

equation for a quadratic function, the equation will have the form

Ex) Tell whether the table of values represents a linear function, an exponential function, or a

quadratic function. Then write an equation for the function.

1) x −2 −1 0 1 2

y 2 0.5 0 0.5 2

2) x −3 −2 −1 0 1

y −7 −5 −3 −1 1

3) x −2 −1 0 1 2

y 8 2 0 2 8

CH 9. Quadratic Equations and Functions - Mr park's...

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