Quadratic Functions

Examples 3x 2 +2x-6 X 2 -4x+3 9x 2 -16

The parabola Graph of a quadratic function is a parabola It’s the “U” shape Upward opening parabola- the coefficient with the x 2 term is positive Has a min value at vertex Domain: Range:

Quadratic Function How do I know it’s a function?

Downward opening parabola- The coefficient with the x 2 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)=x 2 -6x+8 using a table XF(x)=x 2 -6x+8(x, f(x))

Graph f(x)=-x 2 +6x-8 using a table XF(x)=-x 2 -6x-8(x, f(x))

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)=x 2 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)=x 2 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)=x 2, describe the transformation of g(x)=(x+3) 2 +1 and graph the function Shift XY

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

G(x)=x 2 -5 XY

Reflections Reflection across the y axis The function f(x)=x 2 is its own reflection across the y axis F(x)=x 2 G(x)=(-x) 2 =x 2

Reflection across the x axis Function flips across the x axis The entire function gets the negative F(x)=x 2 G(x)=-(x 2 )

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 Ax 2 +Bx+C where a is not equal to zero. Identify a, b, and c – 4x 2 +2x+8 – 2x+9x 2 -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)=-x 2 +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)=x 2 +x-2

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

Analyze the following X 2 -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 -4x 2 -12x-3 Determine if opens up or down, find axis of symmetry, vertex coordinates, y intercept, and graph

Analyze the following -2x 2 -4x Determine if opens up or down, find axis of symmetry, vertex coordinates, y intercept, and graph

Analyze the following X 2 +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=ah 2 +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 List Greatest Common Factor Difference of Squares

Factoring Relay!

Remember Quadratics have the form ax 2 + 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 X 2 +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 x 2 +7x+10

X 2 +12x+32

X 2 +18w+32

X 2 +4x-5

X 2 +10x-24

X 2 -6x-16

X 2 -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

10x 2 -30x

27x 2 +9x-6

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? 4x x x x x 2 -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 – A 2 -b 2 = (a+b)(a-b)

Example 4x 2 -9

100x 2 -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 6x 2 -7x and 6 cannot be added or subtracted to get -7; these are not the factors… TRY ANOTHER

Example 6x 2 -7x and 2 can be added or subtracted to get -7; these are the factors

Example 6x 2 -7x Make up 1 st parentheses Make up second parentheses (2x-3)(3x+1)

Try Another 8x 2 +10x-3

Try Another 9x 2 -15x-14

Try Another 12x 2 +17x-5

The Quadratic Formula

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 Ax 2 +Bx+C=0 Keep in mind you only write out the coefficients, not the x 2 or x

Example Use the quadratic formula to find the roots of x 2 + 5x-14=0

Solve x 2 -7x+6=0

Solve 4x 2 =8-3x

Solve 2x 2 -6x=-3

Practice Use the quadratic formula to solve X 2 +6x=0 X 2 -3x-1=0 X 2 -5x-6=0 4X 2 =-8x-3 5x 2 -2x-3=0 -x 2 -3x+1=0

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