Post on 06-May-2018
CH 9. Quadratic Equations and Functions Algebra I
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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
CH 9. Quadratic Equations and Functions Algebra I
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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 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 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 .
CH 9. Quadratic Equations and Functions Algebra I
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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 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 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 .
CH 9. Quadratic Equations and Functions Algebra I
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Graph
STEP 1 Make a table of values for
.
x 0 2 4
y 4 4
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 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 .
CH 9. Quadratic Equations and Functions Algebra I
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CH 9. Quadratic Equations and Functions Algebra I
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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.
CH 9. Quadratic Equations and Functions Algebra I
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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.
CH 9. Quadratic Equations and Functions Algebra I
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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.
( )
( )
CH 9. Quadratic Equations and Functions Algebra I
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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).
CH 9. Quadratic Equations and Functions Algebra I
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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. ( )
CH 9. Quadratic Equations and Functions Algebra I
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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)
CH 9. Quadratic Equations and Functions Algebra I
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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.
CH 9. Quadratic Equations and Functions Algebra I
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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)
CH 9. Quadratic Equations and Functions Algebra I
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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)
CH 9. Quadratic Equations and Functions Algebra I
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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)
CH 9. Quadratic Equations and Functions Algebra I
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4)
5)
6)
7) –
Ex) Find the number of x-intercepts of the graph of the function.
1)
2)
3)
4)
CH 9. Quadratic Equations and Functions Algebra I
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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
CH 9. Quadratic Equations and Functions Algebra I
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Ex) Use the substitution method to solve the system
1)
2)
3)
4)
CH 9. Quadratic Equations and Functions Algebra I
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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.
(
) (
) (
) ( ) ( )
( ) ( ) ( ) ( ) ( )
CH 9. Quadratic Equations and Functions Algebra I
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( ) ( ) ( ) ( ) ( )
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
CH 9. Quadratic Equations and Functions Algebra I
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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