Chapter 5.1 & 5.2 Quadratic Functions

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

Parts of a Parabola!!! Vertex: highest or lowest point on the graph. 2 ways to find Vertex: 1) Calculator: 2nd  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: 2nd  CALC Zero, Left Bound, Right Bound FIND EACH ONE ON ITS OWN!!

Try Some! Find the vertex and axis of symmetry for each parabola.

Try Some! Find the Vertex, Axis of Symmetry, X-Int and Y-int for each quadratic equation. y = x2 + 2x y = -x2 + 6x + 5 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 We can use systems of equations to write quadratic equations. A = 3 b= -5 and c = 1

The calculator can do it for you! Find a quadratic equation to model the level of water in the water tank. How much water is in the tank after 35 second? When is it empty? STAT  ENTER X-values in L1 and y-values in L2 STAT  CALC 5: QuadReg  ENTER

Chapter 5.3 Translating Parabola

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

Writing a Quadratic Equations

Try one! Write an equations for the following parabola.

Write an equation in vertex form: Vertex (1,2) and y – intercept of 6 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 + c Things 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. y = x2 + 6x – 5 y = 3x2 – 12x + 7 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: 9x2 +3x – 18 7p2 + 21 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 + + - + + - - - Factors (x + _) (x - _) (x - _) ADD SUBTRACT Sign table! Sign + + - + + - - - Factors (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

Slide Factor Divide Reduce More Factoring! When a does NOT equal 1. Steps Slide Factor Divide Reduce

Example! Factor: 1. 3x2 – 16x + 5

Example! Factor: 2. 2x2 + 11x + 12

Example! Factor: 3. 2x2 + 7x – 9

Try Some! Factor 1. 5t2 + 28t + 32 2. 2m2 – 11m + 15