Quadratic Functions and Transformations EQ: What is a quadratic function?

SYMMETRY EQ: What are the different types of symmetry?

ACTIVATION Do the following graphs display symmetry What does symmetry mean?

Lesson Symmetry to a line— X-axis—if their y coordinates are additive inverses Y-axis—if their x coordinates are additive inverses Also called even functions Test if f(-x) = f(x) ie you get the original equation Symmetry to a point– If two points are equidistant from the point of symmetry and all three points are on the same line To the origin if both their x and y coordinates are additive inverses Also called odd functions Test if f(-x) = -f(x) ie ALL the signs change from the original equation

Determine the symmetry of each if it exists: 3y = x x 2 – 2y 2 = 3 Y-axis 3y = (-x) (-x) 2 – 2y 2 =3 3y = x x 2 – 2y 2 = 3 both are symmetric to the y-axis X-axis 3(-y) = x x 2 – 2(-y) 2 =3 -3y = x x 2 – 2y 2 = 3 the second is symmetric to the x-axis Lesson

Determine if the function is even, odd or neither f(x) = 2x 2 + 4xf(x) = 3x 4 – 4x 2 f(-x)= 2(-x) 2 + 4(-x) f(-x) = 3(-x) 4 – 4(-x) 2 f(x) = 2x 2 - 4x f(x) = 3x 4 – 4x 2 Mixed signssame as the original Neithereven Lesson

Do the following graphs display symmetry Lesson

HOMEWORK PAGE(S): 389 NUMBERS: even

TRANSFORMATIONS 9-2 EQ: How do you graph a function of the form f(x)=(x - h) 2 + k??

ACTIVATION How are the parabolas alike and how are they different?

Lesson Transformation— An alteration of the width/scale or direction of a graph Translation— the ability to move a graph left/right and/or up/down

Parent function– the most basic shape of any function These often go through the origin y = x y = x 2 y = x 3 y = |x| y = Lesson

y = a func(bx – h) + k Work with a partner to determine what the h and k do to a function using the following examples y = (x – 2) 2 y = (x + 3) 2 y = x y = x 2 – 4 Lesson Do they do the same thing regardless of the equation? y = |x – 2| y = |x + 3| y = |x| + 2 y = |x| – 4

y = a func(bx – h) + k h moves it left/right +/ – k moves it up/down +/ – Lesson

HOMEWORK PAGE(S): 393 NUMBERS: even

STRETCHING AND SHRINKING 9-3 EQ: How do you graph a function of the form f(x) = a(x) 2 ?

ACTIVATION How are the parabolas alike and how are they different?

y = a func(bx – h) + k Work with a partner to determine what the a does to a function using the following examples y = 2(x ) 2 y = -2(x) 2 y = ½ x 2 y = - ½ x 2 Lesson Do they do the same thing regardless of the equation? y = 2|x| y = -2|x| y = ½ |x| y = - ½ |x|

y = a func(bx – h) + k “a” Determines the width and reflection If ‘a” is positive it is standard (opens up) ie it has a minimum point at the vertex If “a” is negative it is reflected over the y-axis (inverted—opens down) ie it has a minimum point at the vertex If |a|>1 the graph narrows ie 2 makes it goes up twice as fast If |a|<1 the graph is wider ie 1/2 makes it goes up half as fast Lesson

If the graph is f(x) Sketch 3f(x) - ½ f(x) Hint: Pick 0’s and maxs and mins Lesson

HOMEWORK PAGE(S): 398 to 399 NUMBERS: 2 to 8 even

THE ENTIRE TRANSFORMATION 9-4&5 EQ: How do you graph a function of the form f(x) = a(x - h) 2 ? How do you graph a function of the form f(x) = a(x - h) 2 + k?

Lesson Find the vertex, line of symmetry y = 3 (x – 2) Vertex (2, -1) Symmetric to the line x = 2 Opens up so minimum value of y = -1

Lesson Write the equation that is a transformation of f(x) = 2x 2 Given maximum at (3, -1) f(x) = -2(x – 3) 2 – 1

Lesson Given f(x) = the graph to the right Work with a partner to sketch each of the following f(x+1) -2 2f(x) -3f(x – 2) + 3

HOMEWORK PAGE(S): NUMBERS: 38 2 to 16 even 4 to 20 by 4’s

STANDARD FORM FOR QUADRATIC EQUATIONS 9-6 EQ: How do you use standard form of a quadratic equation to solve word problems involving minimum and maximum?

Activation USE ROPES: which means???? R O P E S Read the problem Organize your thoughts in a chart Plan the equations that will work Evaluate the Solution Summarize your findings

Lesson Place each equation in standard form then find the vertex, line of symmetry and max or min value. f(x) = x 2 – 2x – 3 f(x) = x 2 – 2x + f(x) + 4 = (x – 1) 2 f(x) = (x – 1) 2 – 4 11

Lesson Place each equation in standard form then find the vertex, line of symmetry and max or min value. f(x) = 3x 2 – 24x + 50 f(x) – 50 + = 3 (x 2 – 8x + ) f(x) – 2 = 3 (x – 4) 2 f(x) = 3 (x – 4)

A carpenter is building a room with a perimeter of 68 feet. What dimensions would yield the room with maximum area? P = 2x + 2y 68 = 2(x + y) 34 = x + y 34 – x = y Lesson A = lw A = xy A = x (34 – x) A = - x x A + = -1(x 2 – 34x + ) A – 289 = -1(x – 17) 2 A = -1 (x – 17) Vertex (17, 289) Max area 289 sq ft.

HOMEWORK PAGE(S): NUMBERS: 4, 8, 10, 12, 16, 29

GRAPHS AND X-INTERCEPTS 9-7 EQ: How are the x-intercepts of a quadratic function found?

ACTIVATION What is true of an x-intercept?

Lesson x-intercept—the place where a graph crosses the x-axis Means y =0 Find the x-intercept of f(x) = x 2 – 4x = x 2 – 4x + 1 what methods could be used? factoring, completing the square or the quadratic formula

Find the x-intercepts and graph the function If you have the vertex and the x-intercepts you have a good approximation of the graph f(x) = 2x 2 – 4x - 1 Lesson

HOMEWORK PAGE(S): 413 NUMBERS: 2 – 14 even