How do I use intervals of increase and decrease to understand average rates of change of quadratic functions?

Slides:

Advertisements

Similar presentations

SECTION 3.9 SOLVING QUADRATIC INEQUALITIES Feb 25 th 2010 p. 97 in your book.

Advertisements

 Understand that the x-intercepts of a quadratic relation are the solutions to the quadratic equation  Factor a quadratic relation and find its x- intercepts,

Topic 1: Given an equation, determine the type of function represented. Topic 2: Given an equation, describe the function’s rate of change. Topic 3: Given.

Find the solutions. Find the Vertex and Max (-1, 0) (5, 0) (2, 10)

Copyright © Cengage Learning. All rights reserved.

3.2 Quadratic Functions & Graphs

 Find an equation of the tangent line to the curve at the point (2, 1).

EXAMPLE 6 Write a quadratic function in vertex form Write y = x 2 – 10x + 22 in vertex form. Then identify the vertex. y = x 2 – 10x + 22 Write original.

Modeling With Quadratic Functions Section 2.4 beginning on page 76.

4.2 – Graph Quadratic Functions in Vertex or Intercept Form Standard Form: y = ax 2 + bx + c Vertex Form: y = a(x – h) 2 + k.

1.2 Graphing Quadratic Functions In Vertex or Intercept Form

EXAMPLE 3 Graph a quadratic function in intercept form

How do I write quadratic functions and models? 7.2 Write Quadratic Functions and Models Example 1 Write a quadratic function in vertex form Write a quadratic.

Quadratics Functions Review/Notes

EXAMPLE 4 Solve a multi-step problem FOUNTAINS

Solving quadratic equations Factorisation Type 1: No constant term Solve x 2 – 6x = 0 x (x – 6) = 0 x = 0 or x – 6 = 0 Solutions: x = 0 or x = 6 Graph.

9.4 – Solving Quadratic Equations BY GRAPHING!. Warm-Up.

Quadratic Functions A quadratic function is a function with a formula given by the standard form f(x) = ax2+bx+c, where a, b, c, are constants and Some.

FURTHER GRAPHING OF QUADRATIC FUNCTIONS Section 11.6.

Algebra T3 Today: 9.3 Check Up 9.4 Instruction Break Finish 9.4 Practice All Dreams can come true. If we have the courage to pursue them. Walt Disney.

Radical/Power Functions Radicals Complex Numbers Quadratic Functions

Graphs of Quadratic Functions Any quadratic function can be expressed in the form Where a, b, c are real numbers and the graph of any quadratic function.

Definition of a Polynomial Function in x of degree n.

Chapter 2 Polynomial and Rational Functions

How do I graph and use exponential growth and decay functions?

Quadratic Equations and their Applications

Copyright © Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions.

3.1 Quadratic Functions. Polynomials- classified by degree (highest exponent) Degree: 0 -constant function-horizontal line 1 -linear function- 2 -quadratic.

Quadratic Vocabulary Words to graph by….

10.4 Solving Polynomial Equations in Factored Form Objective: I will use the zero-product property to find solutions to polynomial equations that are factored.

How do I graph quadratic functions in vertex and intercept form?

Section 2.6 Quadratic Functions. y = x 2 How many real zeros does it have? How many real zeros can a quadratic function have?

SECTION 3.4 POLYNOMIAL AND RATIONAL INEQUALITIES POLYNOMIAL AND RATIONAL INEQUALITIES.

Notes Over 9.3 Graphs of Quadratic Functions

Chapter 4 Quadratic Functions and Various Nonlinear Topics Section 4.2

Henley Task teaches horizontal transformations Protein Bar Toss Part 1 teaches factoring if a ≠ 1 Section 3.4 for a = 1 Section 3.5 for a ≠ 1 Protein Bar.

3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally.

Essential Question: How do you sketch graphs and write equations of parabolas? Students will write a summary of the steps they use toe sketch a graph and.

Modeling With Quadratic Functions Section 2.4 beginning on page 76.

Aim: How do we apply the quadratic equation? Do Now: Given a equation: a) Find the coordinates of the turning point b) If y = 0, find the values of x.

Warm-Up Exercises Find the product. 1. x + 6 ( ) 3 ANSWER x x

How do graph piecewise functions? 2.5 Use Piecewise Functions Example 1 Evaluate a piecewise function Evaluate the function when x = 3. Solution Because.

Graphing Quadratic Functions

Section 3.1 Day 3 – Quadratic Functions After this section you should be able to: Solve real-world problems using quadratic functions.

Graphing Quadratic Functions. Math Maintenance Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3.

 Given a quadratic equation in standard form, the value of x can be found by using the quadratic formula:

Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions Definition of a polynomial function Let n be a nonnegative integer so n={0,1,2,3…}

Section 4.7: Completing the Square.

Bellwork Identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation. Also describe the Transformations 6

October 28th Happy Birthday tomorrow to: Alex Aviles

Graphing Quadratic Inequalities

Lesson 2.1 Quadratic Functions

2.4 Modeling with Quadratic Functions

Click to see each answer.

Copyright © Cengage Learning. All rights reserved.

Polynomial and Rational Functions

Quadratic Applications

“P. Sherman, 42 Wallaby Way, Sydney!”

Essential Questions How do I use intervals of increase and decrease to understand average rates of change of quadratic functions?

Section 2.1 Quadratic Functions.

“P. Sherman, 42 Wallaby Way, Sydney!”

Copyright © Cengage Learning. All rights reserved.

1. A person throws a baseball into the air with an initial vertical velocity of 30 feet per second and then lets the ball hits the ground. The ball is.

3.3 Interpret Rates of Change of Quadratic Functions p. 69

How do I graph quadratic functions in vertex and intercept form?

5.1 – Rate of Change Textbook pg. 294 Objective:

NO GRAPHING CALCULATORS.

Use Graphs of Functions

Warmup Graph the following quadratic equation using the table provided. Then analyze the graph for the information listed below. y = -3x2 + 12x.

Click to see each answer.

Presentation transcript:

How do I use intervals of increase and decrease to understand average rates of change of quadratic functions?

Example 1 Identify intervals of increase and decrease Graph the function y = x 2 + x – 2. Identify the intervals over which the graph increases and decreases. 3.3 Interpret Rates of Change of Quads Solution You can see from the graph that as you move from left to right the value of the function __________ on the left side of the vertex and __________ on the right side of the vertex. decreases increases The x-coordinate of the vertex is The graph __________ over the interval x > _____ and __________ over the interval x < _____. increases decreases

3.3 Interpret Rates of Change of Quads Checkpoint. Graph the function. Identify the intervals over which the graph increases and decreases.

Example 2 Calculate the average rate of change Calculate the average rate of change of the function y = 2x 2 – 1 on the interval – 1 ≤ x ≤ Interpret Rates of Change of Quads Solution Find the two points on the graph of the function that correspond to the endpoints of the interval. The average rate of change is the slope of the line that passes through these two points. The points are ________ and _________. The average rate of change is:

Example 3 Compare average rates of change Compare the average rates of change of y = 2x 2 and y =  x on 0 ≤ x ≤ Interpret Rates of Change of Quads Solution The average rate of change of y =  x is the slope of the line, which is – 1. y = 2x 2. The average rate of change of y = 2x 2 on 0 ≤ x ≤ 1 is The points ________ and _________ correspond to the endpoints of the interval for y = 2x 2. The average rate of change of y = 2x 2 on 0 ≤ x ≤ 1 is 0 ≤ x ≤ 1. The average rate of change of the quadratic function is ____ times as great as the average rate of change of the linear function on the interval 0 ≤ x ≤ 1.

3.3 Interpret Rates of Change of Quads Checkpoint. Complete the following exercises.

3.3 Interpret Rates of Change of Quads Checkpoint. Complete the following exercises. On y = 2x, it is always 2. 2 is how many times as large as 3/2? The average rate of change of the line is 4/3 times as great as the average rate of change of the quadratic.

Example 4 Solve a real world problem Baseball The path of a baseball thrown at an angle of 40 can be modeled by y =  0.05x x + 8 where x is the horizontal distance (in feet) from the release point and y is the corresponding height (in feet). Find the interval on which the height is increasing. 3.3 Interpret Rates of Change of Quads Solution The height of the baseball will be increasing from the release point until it reaches its maximum height at the vertex. The x-coordinate of the vertex is So the height will be increasing on the interval ________________.

3.3 Interpret Rates of Change of Quads Checkpoint. For the quadratic model in Example 4, find the average rate of change on the given interval.

3.3 Interpret Rates of Change of Quads Pg. 73, 3.3 #1-14 Textbook pg. 73 #2-12 even