QUADRATIC FUNCTIONS CHAPTER 5.1 & 5.2

QUADRATIC FUNCTION A QUADRATIC FUNCTION is a function that can be written in the standard form: f(x) = ax 2 + bx + c where a≠ 0

GRAPHING QUADRATIC The graph of a quadratic function is U- shaped and it is called a PARABOLA. 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: 2 nd  CALC MIN or MAX 2) 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: 2 nd  CALC Zero, Left Bound, Right Bound FIND EACH ONE ON ITS OWN!!

TRY SOME! Find the vertex and axis of symmetry for each parabola.

CALCULATOR COMMANDS Vertex: 2 nd Trace  Min or Max (left bound, right bound, enter) X-Intercepts: 2 nd Trace  Zero (find each one separately) Y-Intercept: 2 nd 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 = x 2 + 2x 2.y = -x 2 + 6x 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  ENTER X-values in L1 and y- values in L2 STAT  CALC 5: QuadReg  ENTER

TRANSLATING PARABOLA 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 – 1 3. y = -3(x + 2) 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 = x 2

VERTEX OF VERTEX FORM

IDENTIFYING THE TRANSLATION Given the following functions, identify the vertex and the translation from y = x 2 1.y = (x + 4) y = -(x – 3) 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 = ax 2 + bx + c Things you will need: a = and Vertex: Vertex: y = a(x – h) 2 + k

EXAMPLE Convert from standard form to vertex form. y = -3x x + 5

EXAMPLE Convert from standard form to vertex form. y = x 2 + 2x + 5

TRY SOME! Convert each quadratic from standard to vertex form. 1.y = x 2 + 6x – 5 2.y = 3x 2 – 12x y = -2x 2 + 4x – 3

WORD PROBLEMS

A ball is thrown in the air. The path of the ball is represented by the equation h = -t 2 + 8t. What does the vertex represent? What does the x-intercept represent?

WORD PROBLEMS A lighting fixture manufacturer has daily production costs of C =.25n 2 – 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: 4x x – 12 EX: 9n 2 – 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.9x 2 +3x – 18 b.7p c.4w 2 + 2w

FACTORS OF QUADRATIC EXPRESSIONS When you multiply 2 binomials: (x + a)(x + b) = x 2 + (a +b)x + (ab) This only works when the coefficient for x 2 is 1.

FINDING FACTORS OF QUADRATIC EXPRESSIONS When a = 1: x 2 + 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 - _) ADDSUBTRACT

EXAMPLES Factor: 1. X 2 + 5x x 2 – 10x x 2 – 6x – x 2 + 4x – 45

EXAMPLES Factor: 1. X 2 + 6x x 2 – 13x x 2 – 5x – x 2 – 16

MORE FACTORING! When a does NOT equal 1. Steps 1.Slide 2.Factor 3.Divide 4.Reduce 5.Slide

EXAMPLE! Factor: 1. 3x 2 – 16x + 5

EXAMPLE! Factor: 2. 2x x + 12

EXAMPLE! Factor: 3. 2x 2 + 7x – 9

TRY SOME! Factor 1. 5t t m 2 – 11m + 15