Entry Task.

Slides:



Advertisements
Similar presentations
Vocabulary axis of symmetry standard form minimum value maximum value.
Advertisements

Objectives Identify quadratic functions and determine whether they have a minimum or maximum. Graph a quadratic function and give its domain and range.
Section 3.6 Quadratic Equations Objectives
THE GRAPH OF A QUADRATIC FUNCTION
Quadratic Functions, Quadratic Expressions, Quadratic Equations
Warm-Up: December 15, 2011  Divide and express the result in standard form.
Solving Quadratic Equations by Graphing
6.6 Finding the Vertex of a Parabola y = a(x – h) + k Vertex: (h, k)
Parts of a Parabola and Vertex Form
Solve Using Best Method
Name:__________ warm-up 9-5 R Use a table of values to graph y = x 2 + 2x – 1. State the domain and range. What are the coordinates of the vertex of the.
FURTHER GRAPHING OF QUADRATIC FUNCTIONS Section 11.6.
Solving Quadratic Equation by Graphing
Review for EOC Algebra. 1) In the quadratic equation x² – x + c = 0, c represents an unknown constant. If x = -4 is one of the solutions to this equation,
Quadratic Functions. Definition of a Quadratic Function  A quadratic function is defined as: f(x) = ax² + bx + c where a, b and c are real numbers and.
1.8 QUADRATIC FUNCTIONS A function f defined by a quadratic equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where c  0, is a quadratic.
© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 8-5 Quadratic Functions, Graphs, and Models.
Graphing Quadratic Functions. Graphs of Quadratic Functions Vertex Axis of symmetry x-intercepts Important features of graphs of parabolas.
Quadratic Functions Objectives: Graph a Quadratic Function using Transformations Identify the Vertex and Axis of Symmetry of a Quadratic Function Graph.
Quiz review Direction of Opening Y – intercept Vertex AOS
Graphing absolute value functions and transformations
9-1 Graphing Quadratic Functions
Warm-Up Find the vertex, the roots or the y- intercept of the following forms: 1. f(x) = (x-4) f(x) = -2(x-3)(x+4) 3. f(x) = x 2 -2x -15 Answers:
Definitions 4/23/2017 Quadratic Equation in standard form is viewed as, ax2 + bx + c = 0, where a ≠ 0 Parabola is a u-shaped graph.
9.1: GRAPHING QUADRATICS ALGEBRA 1. OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form.
Graphing Quadratic Equations Standard Form & Vertex Form.
Solving Quadratic Equations by Graphing Quadratic Equation y = ax 2 + bx + c ax 2 is the quadratic term. bx is the linear term. c is the constant term.
Unit 1B quadratics Day 3. Graphing a Quadratic Function EQ: How do we graph a quadratic function that is in vertex form? M2 Unit 1B: Day 3 Lesson 3.1B.
2.1 – Quadratic Functions.
Chapter 6-1 Graphing Quadratic Functions. Which of the following are quadratic functions?
Solving Quadratic Equations by Graphing 4 Lesson 10.2.
Direction: _____________ Width: ______________ AOS: _________________ Set of corresponding points: _______________ Vertex: _______________ Max or Min?
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally.
Graphs of Quadratic Functions Graph the function. Compare the graph with the graph of Example 1.
4.1 – 4.3 Review. Sketch a graph of the quadratic. y = -(x + 3) Find: Vertex (-3, 5) Axis of symmetry x = -3 y - intercept (0, -4) x - intercepts.
Graphs of Quadratics Let’s start by graphing the parent quadratic function y = x 2.
Quadratic Formula. Solve x 2 + 3x – 4 = 0 This quadratic happens to factor: x 2 + 3x – 4 = (x + 4)(x – 1) = 0 This quadratic happens to factor: x 2.
Fri 12/11 Lesson 4 – 1 Learning Objective: To graph quadratic functions Hw: Graphing Parabolas Day 1 WS.
Quadratic Functions Solving by Graphing Quadratic Function Standard Form: f(x) = ax 2 + bx + c.
For the function below, find the direction of opening, the equation for the axis of symmetry, and the y-intercept. Use this information to sketch the.
CHAPTER 10 LESSON OBJECTIVES. Objectives 10.1 Students will be able to: Identify quadratic functions and determine whether they have a minimum or maximum.
Solving Quadratic Equation by Graphing Students will be able to graph quadratic functions.
Section 3.1 Day 2 – Quadratic Functions After this section you should be able to: Graph a quadratic function with and without a calculator. Find the coordinates.
Graphing Quadratic Functions. The graph of any Quadratic Function is a Parabola To graph a quadratic Function always find the following: y-intercept.
Unit 10 – Quadratic Functions Topic: Characteristics of Quadratic Functions.
Textbook Chapter 3 and 4 Math 20-1 Chapter 3 Quadratic Functions 3.1 B Quadratic Function in Vertex Form Teacher Notes 2.
Quadratic Functions Sections Quadratic Functions: 8.1 A quadratic function is a function that can be written in standard form: y = ax 2 + bx.
Identifying Quadratic Functions. The function y = x 2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function.
Characteristics of Quadratic Functions CA 21.0, 23.0.
Entry Task. Take a look…. y = x(18-x) Then we had y = -x 2 +18x We could graph this using symmetry and find the zero’s. if x is 0 what is y? 0 or 18.
Solving Quadratic Equations by Graphing Need Graph Paper!!! Objective: 1)To write functions in quadratic form 2)To graph quadratic functions 3)To solve.
Lesson 8-1 :Identifying Quadratic Functions Lesson 8-2 Characteristics of Quadratic Functions Obj: The student will be able to 1) Identify quadratic functions.
Factor each polynomial.
Quadratic Functions, Quadratic Expressions, Quadratic Equations
Warm Up /05/17 1. Evaluate x2 + 5x for x = -4 and x = 3. __; ___
Warm Up /31/17 1. Evaluate x2 + 5x for x = 4 and x = –3. __; ___
Quadratic Equations Chapter 5.
Mrs. Rivas Ch 4 Test Review 1.
Warm-Up Find the x and y intercepts: 1. f(x) = (x-4)2-1
Graphing Quadratic Functions
Y Label each of the components of the parabola A: ________________ B: ________________ C: ________________ C B B 1 2.
THE VERTEX OF A PARABOLA
Unit 12 Review.
Entry Task.
Bellwork: 2/6/18 2) Factor: x2-x-6 (x-6) (2x+5)
4.1 Notes – Graph Quadratic Functions in Standard Form
Algebra 2 – Chapter 6 Review
Warm Up.
Quadratic Equation Day 4
Graphing Quadratic Functions
Presentation transcript:

Entry Task

4.1 Graphing Quadratic Functions Learning Target: I can Identify and graph quadratic equations.

Graphs of Quadratic Functions Axis of symmetry Important features of graphs of parabolas Maximum x-intercepts Minimum Vertex

Graphing Quadratics If you were asked to graph a quadratic, what information would you need to know to complete the problem? The vertex, because we need to know where the graph is located in the plane If the parabola points up or down, and whether it opens normal, narrow or wide Our graphs will be more “quick sketches” than exact graphs.

Graph of f(x)=x2 Axis of symmetry is x = 0 x f(x) 1 -1 2 -2 4 1 -1 2 -2 4 Points up, opens “normal” Notice the symmetry Vertex at (0, 0) The “mother function”

More with Vertex Form The vertex is (h, k). Changes in (h, k) will shift the quadratic around in the plane (left/right, up/down). The axis of symmetry is x = h If a > 0, the graph points up If a < 0, the graph points down Example #2 Example #1 Vertex is (4, 0) Axis is x = 4 Points down Vertex is _____ Axis is _______ Points _______ Vertex is (0, 6) Axis is x = 0 Points up Vertex is _____ Axis is _______ Points _______ Example #3 Vertex is (-3, -1) Axis is x = -3 Points up Vertex is _____ Axis is _______ Points _______ Notice that you take the opposite of h from how it is written in the equation

Equations of Quadratic Functions Vertex Form Standard Form

More about a When a = 1, the graph is “normal” a =1 a =1/5 a = 5 What happens to the graph as the value of a changes? If a is close to 0, the graph opens _______________ If a is farther from 0, the graph opens ____________ If a > 0, the graph points________ If a < 0, the graph points ________ If a is close to 0, the graph opens wider If a is farther from 0, the graph opens narrower If a > 0, the graph points up If a < 0, the graph points down

Max and Min Problems What is the definition of the maximum or minimum point of a quadratic function? The vertex of a quadratic function is either a maximum point or a minimum point max min If a quadratic points down, the vertex is a maximum point If a quadratic points up, the vertex is a minimum point If you are asked to find a maximum or minimum value of a quadratic function, all you need to do is find its vertex

Sketch each quadratic Identify the vertex, axis of symmetry, the max or min value, and domain and range. AOS: x = -3 Min: k =-1 D: all reals R: all reals > - 1 V = (-3, -1) Points up Narrow AOS: x = 0 Max: k = 4 D: all reals R: all reals < 4 V = (0, 4) Points down Wide

Sketch each quadratic V = (-3, -1) V = (2, 1) Points up Points down Narrow V = (2, 1) Points down Normal V = (-4, 2) Points up Normal V = (0, 4) Points down Wide

Homework Homework – p. 199 #9-36 by 3’s Challenge - 56

Summary: Be able to compare and contrast vertex and standard form   Vertex Form Standard Form How do you find the Vertex? How do you find the Axis of Symmetry? How can you tell if the function: points up or down? opens normal, wide or narrow? What info is needed to do a quick sketch or graph? How do you find the solutions? (x-intercepts, roots, zeroes, value of x when y = 0) Set = 0, get “squared stuff” alone, then use square root method Set = 0 and use method of choice (factor, formula or square root)

More with Standard Form To find the x-value of the vertex, use the formula To find the y-value, plug in x and solve for y The axis of symmetry is If a > 0, the graph points up If a < 0, the graph points down Example #1 b = 4, a = -1 Find x-value of vertex using formula Vertex is _____ Axis is _______ Points _______ Vertex is (2, 1) Axis is x = 2 Points down Find y-value using substitution 2 (2) (2)

More examples Example #2 You try: Vertex is _____ Axis is _______ Points _______ Find x-value using formula b = -1, a = 3 Find y-value using substitution 16 16 16 Vertex is (3, 17) Axis is x = 3 Points down Vertex is _____ Axis is _______ Points _______ Vertex is (1/6, 59/12) Axis is x = 1/6 Points up

Finding x-intercepts of quadratic functions What are other words for x-intercepts? Name 4 methods of finding the x-intercepts of quadratic equations: roots zeroes solutions All are the value of x when y = 0 factoring The Square Root The Quadratic Formula Graphing

Example An object is thrown upward from the top of a 100 foot cliff. Its height in feet about the ground after t seconds given by the function f(t) = -16t2 + 8t + 100. What was the maximum height of the object? How many seconds did it take for the object to reach its max height? How can we find the answer? What is the question asking for?

vertex Example What was the maximum height of the object? How many seconds did it take for the object to reach its maximum height? What is the definition of the maximum or minimum point of a quadratic function? The vertex of a quadratic function is either a maximum point or a minimum point vertex

Example f(t) = -16t2 + 8t + 100. f(1/4) = -16(1/4)2 + 8(1/4) + 100. Step 2: Understand the equation Example y x Input: time Output: height Step 1: Visualize the problem f(t) = -16t2 + 8t + 100. To find the max values, find the vertex The x-value of the vertex is the max time (1/4, 101) It took about .25 seconds for the object to reach its max height The y-value of the vertex is the max height f(1/4) = -16(1/4)2 + 8(1/4) + 100. f(t) = -16t2 + 8t + 100. The max height was 101 feet f(1/4) = 101