Graphing Quadratic Functions

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Presentation transcript:

Graphing Quadratic Functions Chapter 5 Section 1 Graphing 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 Form What 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:

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 20-25 DIRECTIONS: find the vertex only!!!

How to graph a quadratic function Step 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

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 2 Warm-Up Review Homework Recap Word Problems and partner activity Notes on graphing in Intercept Form Group Activity

Warm up: y = -½x2 + 4x - 4 X Y 3 3.5 4 5

Homework Review http://www.kutasoftware.com/FreeWorksheets/A lg2Worksheets/Properties%20of%20Parabolas. pdf

Notes… Real-life application The 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). Find the engine speed that maximizes torque. 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 Down Wide 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 Form Review 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 Form Steps: 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 Form Steps: 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 = 0 Plug 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 notecards Within 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/M2 C05AGD.PDF