Q U A D RAT I C F U N C T I O N S

CHAPTER 5.1 & 5.2

QUADRATIC FUNCTION

A QUADRATIC FUNCTION is a function that can be written in the standard form:

f(x) = ax2 + bx + c where a≠ 0

GRAPHING QUADRATIC

The graph of a quadratic function is U-shaped and it is called a PARABOLA.

a < 0 a > 0

Parabola opens down

Parabola

opens up

PARTS OF A PARABOLA

Vertex: highest or lowest point on the graph.

2 ways to find Vertex:

1) Calculator: 2nd CALCMIN or MAX2) Algebraically

PARTS OF A PARABOLA

Axis of symmetry: vertical line that cuts the parabola in half

Always x = a Where a is the x from the vertex

PARTS OF A PARABOLA!!!

Corresponding Points: Two points that are mirror images of each other over the axis of symmetry.

PARTS OF A PARABOLA!!!

Y-intercept:Where the parabola crosses the Y-Axis.

To find:Look at the table where x is zero.

PARTS OF A PARABOLA!!!

X- Intercept: The the parabola cross the x-axis.

To find:2nd CALCZero,Left Bound, Right BoundFIND EACH ONE ON ITS OWN!!

TRY SOME!

Find the vertex and axis of symmetry for each parabola.

CALCULATOR COMMANDS

Vertex: 2nd Trace Min or Max(left bound, right bound, enter)

X-Intercepts: 2nd Trace Zero(find each one separately)

Y-Intercept: 2nd Graph find where x is zero

(or trace and find where x is zero on the graph)

TRY SOME!

Find the Vertex, Axis of Symmetry, X-Int and Y-int for each quadratic equation.1. y = x2 + 2x

2. y = -x2 + 6x + 5

3. y = ¼ (x + 5)2 – 3

TRY SOME!

Identify the vertex of the graphs below, the axis of symmetry and the points that correspond with points P and Q.

WRITING QUADRATIC EQUATIONS

Quadratic Regression

STAT ENTERX-values in L1 and y-values in L2STAT CALC5: QuadReg ENTER

T RA N S L AT I N G PA RA BO L A

CHAPTER 5.3

STANDARD FORM VERTEX

VERTEX FORM

Graph the following functions. Identify the vertex of each.

1. y = (x – 2)2

2. y = (x + 3)2 – 13. y = -3(x + 2)2 + 44. y = 2(x + 3)2 + 1

VERTEX OF VERTEX FORM

The Vertex form of a quadratic equation is a translation of the parent function y = x2

VERTEX OF VERTEX FORM

IDENTIFYING THE TRANSLATION

Given the following functions, identify the vertex and the translation from y = x2

1. y = (x + 4)2 + 7 2. y = -(x – 3)2 + 13. y = ½ (x + 1)2

4. y = 3(x – 2)2 – 2

WRITING A QUADRATIC EQUATIONS

TRY ONE!

Write an equations for the following parabola.

ONE MORE!

Write an equation in vertex form:Vertex (1,2) and y – intercept of 6

CONVERTING FROM STANDARD TO VERTEX FORM

Things needed:

Find Vertex using x = -b/2a, and y = f(-b/2a) This is your h and k.Then use the the a from standard form.

CONVERTING FROM STANDARD TO VERTEX

Standard: y = ax2 + bx + cThings you will need:

a = and Vertex:

Vertex: y = a(x – h)2 + k

EXAMPLE

Convert from standard form to vertex form.

y = -3x2 + 12x + 5

EXAMPLE

Convert from standard form to vertex form.

y = x2 + 2x + 5

TRY SOME!

Convert each quadratic from standard to vertex form.1. y = x2 + 6x – 5

2. y = 3x2 – 12x + 7

3. y = -2x2 + 4x – 3

WORD PROBLEMS

WORD PROBLEMS

A ball is thrown in the air. The path of the ball is represented by the equation h = -t2 + 8t. What does the vertex represent?What does the x-intercept represent?

WORD PROBLEMS

A lighting fixture manufacturer has daily production costs of C = .25n2 – 10n + 800, where C is the total daily cost in dollars and n is the number of light fixture produced. How many fixtures should be produced to yield minimum cost.

FACTORING

GCF

One way to factor an expression is to factor out a GCF or a GREATEST COMMON FACTOR.

EX: 4x2 + 20x – 12

EX: 9n2 – 24n

FACTORS

Factors are numbers or expressions that you multiply to get another number or expression.

Ex. 3 and 4 are factors of 12 because 3x4 = 12

FACTORS

What are the following expressions factors of?

1. 4 and 5? 2. 5 and (x + 10)

3. 4 and (2x + 3) 4. (x + 3) and (x - 4)

5. (x + 2) and (x + 4) 6. (x – 4) and (x – 5)

TRY SOME!

Factor:a. 9x2 +3x – 18

b. 7p2 + 21

c. 4w2 + 2w

FACTORS OF QUADRATIC EXPRESSIONS

When you multiply 2 binomials:

(x + a)(x + b) = x2 + (a +b)x + (ab)

This only works when the coefficient for x2 is 1.

FINDING FACTORS OF QUADRATIC EXPRESSIONS

When a = 1: x2 + bx + c

Step 1. Determine the signs of the factors

Step 2. Find 2 numbers that’s product is c, and who’s sum is b.

SIGN TABLE!

Sign + + - + + - - -

Factors

(x + _)

(x + _)

(x - _)(x - _)

(x + _)(x - _)

(x + _)(x - _)

ADD SUBTRACT

EXAMPLES

Factor:1. X2 + 5x + 6 2. x2 – 10x + 25

3. x2 – 6x – 16 4. x2 + 4x – 45

EXAMPLES

Factor:1. X2 + 6x + 9 2. x2 – 13x + 42

3. x2 – 5x – 66 4. x2 – 16

MORE FACTORING!

When a does NOT equal 1.Steps

1. Slide2. Factor3. Divide4. Reduce5. Slide

EXAMPLE!

Factor:1. 3x2 – 16x + 5

EXAMPLE!

Factor:2. 2x2 + 11x + 12

EXAMPLE!

Factor:3. 2x2 + 7x – 9

TRY SOME!

Factor1. 5t2 + 28t + 32 2. 2m2 – 11m + 15