Chapter 5 Section 1 Graphing Quadratic Functions

Chapter 5Section 1Graphing Quadratic Functions

Graphing Quadratic Functions (just read)Objective:

You will be given 2 forms of quadratic functions.

Based on the form, you will learn how to graph a quadratic function.

What is a Quadratic Function?Definition Graph

• Has an “x2” in the equation

• Takes the shape of a “U” called a parabola

• Can “smile” or “frown”

• The lowest or highest point is the vertex

Form #1: Standard FormWhat it looks like The impact of a

y = ax2 + bx + c

a, b, and c are numbers

a cannot equal zero

• If a is positive, then the parabola opens upwards (smile)

• If a is negative, then the parabola opens downwards (frown)

• If a is a fraction, parabola will be “wide”

• If a is a whole number not equal to 1, parabola will be “skinny”

Let’s look at a few examples on the graphing calculator

y = x2 + 3x – 1 y = -x2 + x – 4

Let’s look at a few examples on the graphing calculator

y = ½x2 + x – 1 y = -¾x2 – 4

y = 3x2 + x – 3 y = -2x2 + 2x - 5

Vertex:

a

bx

2

Vertex

To find the vertex:Step 1: write down a and b Step 2: plug into formula and get value – THIS IS THE X-COORDINATE!!!Step 3: take that value and plug into equation to get y

Now you have the vertex (x, y)

Example

Find the vertex of y = 2x2 – 8x + 6.a =

b =

x =

Let’s go our book…..

Page 253 Complete 20-25DIRECTIONS: find the vertex only!!!

How to graph a quadratic functionStep 1: Write down the characteristics of

the function.Step 2: Find the vertex. Plot it!Step 3: Choose two more x-values one more

and one less than the vertex.

Example:

X Y

32 2 xy

Practice on your own

•Let’s graph 20 – 24. •Use the vertex you found and plot.•Find two more points and plot.

Quadratic Functions Day 2Warm-UpReview HomeworkRecapWord Problems and partner activityNotes on graphing in Intercept FormGroup Activity

Warm up: y = -½x2 + 4x - 4

X Y

3 3.5

4 4

5 3.5

Homework Review

http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Properties%20of%20Parabolas.pdf

Notes…Real-life applicationThe engine torque y (in foot-pounds) of one

model of car is given by y = -3.75x2 + 23.2x + 38.8

Where x is the speed of the engine (in thousands of revs per minute).

1.Find the engine speed that maximizes torque.

2.What is the maximum torque?(revs per minute, torque)

What are the characteristics of this quadratic?The engine torque y (in foot-pounds) of one

model of car is given by y = -3.75x2 + 23.2x + 38.8

Faces DownWide

This is why the question asks for the maximum. The vertex will give:

x = max revolutions, and y = max torque.

Real-Life Application continued

Use the vertex formula to answer the questions…

a = -3.75

b = 23.2

x =

Partner Activity

•Choose a partner•1 person is the writer•1 person is the reported (to me)•You need a piece of paper •Complete page 255 #55 a and b only!!!•Round your answers to the nearest

whole number

Recap:

•A parabola that “smiles” or faces upwards (where a is a positive number) it will have a minimum.

•A parabola that “frowns” or faces downwards (where a is a positive number) it will have a maximum.

Form #2: Intercept FormReview of Intercepts Characteristics of quadratics

Recall: An intercept is where are graph crosses the x or y axis

(x, 0) is the point – this is called the x-intercept

• Parabolas have either none, one, or two x-intercepts

• Intercept form looks like a factored equation

• Intercept form looks like: y = a(x – p)(x – q)

Graphing in Intercept FormSteps: Ex: y = (x - 2)(x + 2)

Set each binomial equal to zero and solve for x. These will give you the intercepts.

x - 2 = 0 x + 2 = 0 + 2 +2 - 2 -2_______ _________x = 2 x = -2(2, 0) (-2, 0)

Graphing in Intercept FormSteps: Ex: y = (x - 2)(x + 2)

Plot these points.The vertex is halfway between

these points. To find this: (p + q)/2 This is the x-

coordinate of the vertex!(-2 + 2)/2 = o/2 = 0Plug in x-coordinate to get y. y = (0 – 2)(0 + 2) y = (-2)(2) y = -4 so (0, -4)Plot vertex and draw the

parabola

Let’s try together…..y = -(x – 3)(x + 1) y = 2x(x – 4)

Group Assignment on notecardsWithin your group Assignment

One person is the writer/recorder

One person is the reporter

All work together on answering the questions!!!

• You are given 3 notecards

• Each problem needs to be on graph paper

• All three points must be labeled (intercepts and vertex)

Homework

•Page 254 32, 33, 34, 36

http://olp.classzone.com/materials/2612962/M2C05AGD.PDF