Lesson 5-1: Modeling Data with Quadratic Functions--identify quadratic functions & their graph --Quadratic regression on graphing

calculator

Example: Find the quadratic model for the values (-1, 10), (2, 4), (3, -6)

The quadratic function in standard form is ____________________________________

Lesson 5-2: Properties of Parabolas--graphing from standard form --finding minimum and maximum values of quadratic

functions

ALL GRAPHS SHOULD INCLUDE: --axis of symmetry --vertex --two pts to left and right of vertex

STANDARD FORM OF QUADRATIC FUNCTION: y = ax2 + bx + cQuadratic term: ax2 Linear term: bx Constant term: c

Example: Determine if the function is linear or quadratic. State the quadratic, linear, and constant terms:1) f(x) = 3x2 – (x + 3)(2x – 1)

GRAPH OF A QUADRATIC FUNCTION:Example: 1) State vertex:2) State axis of symmetry:3) State P’4) State Q’

P

Q

Graph: (y = ax2 + c)Ex: y = -x2 + 4

Graph: (y = ax2 + bx + c)Ex: y = x2 + 2x - 6

Parabola – the graph of a quadratic function (U-shaped)Axis of Symmetry -- the line that divides a parabola into two parts that are mirror images.Vertex – the point at which the parabola intersects the axis of symmetry. If graph opens up, vertex is MINIMUM. If graph opens down, vertex is MAXIMUM.

GRAPH FROM STANDARD FORM: f(x) = ax2 + bx + c:

1) If a is (+), parabola opens up.2) If a is (-), parabola opens down.3) Vertex: (4) Axis of symmetry: x = 5) Y-intercept: (0, c)

QUADRATIC REGRESSION ON GRAPHING CALCULATOR: Write a quadratic model given points on the graph. In calculator:

1) STAT / 1:Edit – enter x values in L1 and y values in L2 2) STAT / CALC / 5:QuadReg 3) VARS / Y-VARS / 1:Function / 1:Y1 4) ENTERSubstitute values of a, b, and c into standard form.

CH 5 PT 1 PORTFOLIO PAGE – 5-1 THROUGH 5-5

M. MURRAY

f(x) = x2 – 5x + 3; quadratic/ x2 / -5x / 3

(4, 1)X = 4(2, 0)(8, -3)

Y = -2x2 + 12

V: (0, 4)Pts: (1, 3);(2, 0)

V: (-1, -7))Pts: (0, -6);(1, -3)

Lesson 5-3: Translating Parabolas--Using Vertex Form to graph and write equations

VERTEX FORM OF QUADRATIC FUNCTION: y = a(x – h)2 + k

WRITING EQUATIONS: Given vertex and one pointExample: Vertex: (-2, 5) Point: (3, 4)

IDENTIFY VERTEX AND Y-INTERCEPT FOR EACH FUNCTION:Example (standard form) Example (vertex form)Y = -3x2 + 6x – 1 y = (x – 2)2 + 3

STANDARD FORM/VERTEX FORM:Convert the function to standard form: Convert the function to vertex form: y = 2(x – 3)2 + 4 y = 2x2 – 4x + 3

Lesson 5-5: Solving Quadratic Equations --by factoring, graphing, and square roots

Vertex: (h, k) Axis of Symmetry: x = h

Graph: y = -3(x+1)2 + 4

TO SOLVE QUADRATIC EQUATIONS BY FACTORING: 1) Write equations in standard form (set = to zero)2) Factor3) Apply zero product property and set each variable factor to zero.4) Solve the equations

TO SOLVE BY FINDING SQUARE ROOTS: 1) Isolate squared term on one side of equation2) Take the square root of each side. *don’t forget

1) x2 = 16x – 48 2) 9x2 – 16 = 0

3) x2 – 5x + 2 = 0TO SOLVE BY GRAPHING:

 1) Graph the related function y = ax2 + bx + c2) Find ZEROS (x-intercepts):

2nd/CALC/Zero Left bound, Right bound, Guess M. MURRAY

V: (-1, 4)Pts: (0, 1);(1, -8)

Y = -

Y = 2x2 + 6x + 4

V: (2, 3)Y-int: (0, 7)V: (1, 2)

Y-int: (0, -1)

Y = 2(x – 1)2 + 1

x = 12, x = 4

x =