Quadratic Functions

Examples

•3x2+2x-6•X2-4x+3•9x2-16

The parabola• Graph of a quadratic function is a parabola• It’s the “U” shape• Upward opening parabola- the coefficient

with the x2 term is positive• Has a min value at vertexDomain:Range:

Quadratic Function

• How do I know it’s a function?

• Downward opening parabola- The coefficient with the x2 term is negative

• Has a max value at the vertex• Domain: • Range:

Plotting Quadratics

• You can graph a quadratic function by plotting points with coordinates that make the equation true

• Plug in numbers for the x value and simplify

Graph f(x)=x2-6x+8 using a tableX F(x)=x2-6x+8 (x, f(x))1

2

3

4

5

Graph f(x)=-x2+6x-8 using a tableX F(x)=-x2-6x-8 (x, f(x))1

2

3

4

5

Translating Quadratic Functions

• They’re BAAAACCCCKKKKK!!!!• Vertex Form

Y=a(x-h)2+k

Stretch or Compression factor

Horizontal shift

Vertical Shift

Horizontal Shifts

• F(x)=x2

• Shift to the right g(x)=(x-h)2

• Shift to the left g(x)=(x+h)2

• Again, opposite signs

Vertical Shifts

• F(x)=x2

• Vertical Shift Up g(x)=f(x)+k• Vertical Shift Down g(x)=f(x)-k

Practice with Shifts

• Using the graph of f(x)=x2, describe the transformation of g(x)=(x+3)2+1 and graph the function

• Shift X Y-2-1012

G(x)=(x-2)2-1

X Y-2-1012

G(x)=x2-5

X Y-2-1012

Reflections

• Reflection across the y axis• The function f(x)=x2

is its own reflection across

the y axis• F(x)=x2

• G(x)=(-x)2=x2

Reflection across the x axis

• Function flips across the x axis• The entire function gets the negative• F(x)=x2

• G(x)=-(x2)

Horizontal Stretch/Compression

• Remember, take the reciprocal of the stretch, compression factor

• Changes only the number in front of the x

Vertical Stretch/Compression

• Do not take the reciprocal• Changes the output of the function

Preview of New Vocab

Can you identify the symmetry?

Maximum and Minimum

• Related words?

Symmetry

• Parabolas are symmetric curves– Reflection over the y axis results in same function

• Axis of symmetry- line through the vertex of a parabola that divides the parabola into two identical halves

• Quadratic in vertex form has axis of symmetry x=h (horizontal shift)

Identify axis of symmetry

• What kind of line do you think the axis of symmetry is?

• F(x)=2(x+2)2-3

• F(x)=(x-3)2+1

Standard Form

Standard Quadratic Form

• Any function that can be written in the form Ax2+Bx+C where a is not equal to zero.

• Identify a, b, and c– 4x2+2x+8– 2x+9x2-4

What it can tell you

• Leading coefficient in standard form: What is it?

• If the leading coefficient is >1, opens up• If leading coefficient is <1, opens down

a>1

• Domain:

• Range:

• Maximum or Minimum value at the vertex?

Smiley face activity

How to find things in standard form?

• Vertical stretch or compression: look at leading coefficient

• Opens up or down: look at leading coefficient

Axis of Symmetry

• What is the axis of symmetry?• What type of line would run through the axis

of symmetry• h=-b/2a• Note: The – means that you make the b value

opposite what it is in the function• If it is negative to begin with, make it positive,

vice versa

Find the axis of symmetry

• F(x)=-x2+4• Write down the values of a, b, and c

Finding the vertex

• If the vertex is the highest or lowest POINT on a parabola, it makes sense that it would be written as an _____________________

• The vertex lies on the axis of symmetry

Finding the Vertex

• Find the axis of symmetry, this is the x coordinate of the vertex

• Plug the x value into the original function• Solve for y• This is the y coordinate of the vertex• THE VERTEX IS ALSO THE MAXIMUM OR

MINIMUM!!!!!

Find the vertex

• F(x)=x2+x-2

Finding the y intercept

• This one is easy!!• It’s the value of the constant!• Example: Find the y intercept: f(x)=4x2+2x-8• Don’t believe me? Graph it!

Analyze the following

• X2-4x+6• Determine if opens up or down, find axis of

symmetry, vertex coordinates, y intercept, and graph, and the maximum or minimum

Analyze the following

• -4x2-12x-3• Determine if opens up or down, find axis of

symmetry, vertex coordinates, y intercept, and graph

Analyze the following

• -2x2-4x• Determine if opens up or down, find axis of

symmetry, vertex coordinates, y intercept, and graph

Analyze the following

• X2+3x-1• Determine if opens up or down, find axis of

symmetry, vertex coordinates, y intercept, and graph

Pairs Practice

• Work in your groups of two• Work separately helping one another when

needed• When the problem is circled, you are to stop

and check your answer with your partner• If your answers match, AWESOME.• If not, try to figure out where you went

wrong!

Graphic Organizer!

Standard Form

• Coefficients a, b, and c can show the properties of the graph of the function

• You can determine these properties by expanding the vertex form

• a(x-h)2+k

Standard and Vertex Equivalents

• a; same as in vertex form– Indicates a reflection and/or vertical stretch,

compression

• b=-2ah– Solving for h gives the axis of symmetry– h=-b/2a

• c=ah2+k– c is the y intercept

Venn Diagram

Summary

• Parabola opens up if a>0• Parabola opens down if a<0• Axis of symmetry is a vertical line =-b/2a• Vertex: (-b/2a, f(-b/2a))

Factoring

Factor ListGreatest Common Factor

Difference of Squares

Factoring Relay!

Remember

• Quadratics have the form ax2 + bx+c• Sometime we need to factor them to see their

solutions• Factoring, setting the equation equal to zero

and solving for x allows us to find the x intercepts

• X intercepts also called the zeros or the roots

Zeros

• A zero of a function is a value of the input x that makes the output of f(x)=0

• Quadratic functions can have two zeros which are always symmetric about the axis of symmetry

Find the zeros

• X2+2x-3 using a graph and table• Find the vertex and plot using a table• Double check by graphing in calculator

Another Way

• Zero product property• Set the equation equal to zero, and factor• Once factored, set each set of parentheses

equal to zero and solve for x

Factor Tree Method

• Works for quadratics with leading coefficients (A values) of 1

• Does not matter if the quadratic has positive or negative b and c terms

Factor Tree Method

• Take the last term of your quadratic (C value) and list all of the possible combinations that will multiply to give you that number

• Look for the combination that adds or subtracts to give you the middle term, the b value

Example

• Factor x2+7x+10

• X2+12x+32

• X2+18w+32

• X2+4x-5

• X2+10x-24

• X2-6x-16

• X2-35x+34

Greatest Common Factor

GCF

• Can be done with any number of terms• Goal is to find what numbers or variables or

numbers and variables are common to ALL the terms

• Once you find the GCF, divide each term by the GCF

Example

• 6x+12

• 10x2-30x

• 27x2+9x-6

Difference of Squares

Difference of Squares

• Works only when…– You have two terms (A and B)– Must be separated by a minus sign– Both terms are perfect squares

Will Difference of Squares Work?

• 4x2-16• 4x2+16• 9x2+16• 15x2-25• 25x2-49

Difference of Squares

• If you meet all the conditions, you write out two sets of parentheses

• ( )( )• Take the square root of each term and place

in each parentheses• Give one parentheses a + sign and the other a

–• A2-b2= (a+b)(a-b)

Example

• 4x2-9

• 100x2-81

Warm up!

• Explain in words how you use the Factor List method to factor, and when it can be used. Give an example

Factoring

Leading Coefficient is not equal to 1

Method of determining what goes in parentheses

• Create a table (s)

• Cross multiply• The cross multiplied terms must add or

subtract to give middle term if correct

Possible factors of a

Possible factors of b

Example

• 6x2-7x-3

6 1

3 1

3 6

3 and 6 cannot be added or subtracted to get -7; these are not the factors… TRY ANOTHER

Example

• 6x2-7x-3

2 3

3 1

9 29 and 2 can be added or subtracted to get -7; these are the factors

Example

• 6x2-7x-3

2 3

3 1

9 2Make up 1st parentheses Make up second parentheses

(2x-3)(3x+1)

Try Another

• 8x2+10x-3

Try Another

• 9x2-15x-14

Try Another

• 12x2+17x-5

The Quadratic Formula

a

acbb

2

42

Why use it?

• Real world applications of quadratics and parabolic motion are not always solved through factoring

• Quadratic lets you solve the problem whether it is factorable or not.

What do the variables mean?

• They represent the coefficients from the quadratic expression

• Ax2+Bx+C=0• Keep in mind you only write out the

coefficients, not the x2 or x

Example

• Use the quadratic formula to find the roots of x2 + 5x-14=0

• Solve x2-7x+6=0

• Solve 4x2=8-3x

• Solve 2x2-6x=-3

Practice

• Use the quadratic formula to solve• X2+6x=0• X2-3x-1=0• X2-5x-6=0• 4X2=-8x-3• 5x2-2x-3=0• -x2-3x+1=0

Challenge

• Solve using the quadratic formula• (x-4)(x+5)=7