Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction Previously, we calculated the rate of change of linear functions as well as intervals of exponential functions. We observed that the rate.

Published byClaud Watts Modified over 8 years ago

Similar presentations

Presentation on theme: "Introduction Previously, we calculated the rate of change of linear functions as well as intervals of exponential functions. We observed that the rate."— Presentation transcript:

1 Introduction Previously, we calculated the rate of change of linear functions as well as intervals of exponential functions. We observed that the rate of change of exponential functions changed if we focused on different intervals of the function. Here, we will focus on graphs of linear and exponential functions and learn how to estimate the rates of change. 1 3.3.3: Recognizing Average Rate of Change

2 Key Concepts To determine the rate of change of a function, first identify the coordinates of the interval being observed. Sometimes it is necessary to estimate the values for y. The resulting calculation may be an estimation of the rate of change for the interval identified for the given function. 2 3.3.3: Recognizing Average Rate of Change

3 Key Concepts, continued 3 3.3.3: Recognizing Average Rate of Change Estimating Rate of Change from a Graph 1.Determine the interval to be observed. 2.Identify (x 1, y 1 ) as the starting point of the interval. 3.Identify (x 2, y 2 ) as the ending point of the interval. 4.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. 5.The result is the estimated rate of change for the interval between the two points identified.

4 Common Errors/Misconceptions incorrectly choosing the values of the indicated interval to estimate the rate of change incorrectly estimating the values for the indicated interval substituting incorrect values into the formula for calculating rate of change 4 3.3.3: Recognizing Average Rate of Change

5 Guided Practice Example 2 Jasper has invested an amount of money into a savings account. The graph to the right shows the value of his investment over a period of time. What is the rate of change for the interval [1, 3]? 5 3.3.3: Recognizing Average Rate of Change

6 Guided Practice: Example 2, continued 1.Determine the interval to be observed. The interval to observe is [1, 3], or where 1 ≤ x ≤ 3. 6 3.3.3: Recognizing Average Rate of Change

7 Guided Practice: Example 2, continued 2.Identify the starting point of the interval. The x-value of the starting point is 1. The corresponding y-value is approximately 550. The starting point of the interval is (1, 550). 7 3.3.3: Recognizing Average Rate of Change

8 Guided Practice: Example 2, continued 3.Identify the ending point of the interval. The x-value for the ending point is 3. The corresponding y-value is approximately 1,100. The ending point of the interval is (3, 1100). 8 3.3.3: Recognizing Average Rate of Change

9 Guided Practice: Example 2, continued 9 3.3.3: Recognizing Average Rate of Change

10 Guided Practice: Example 2, continued 4.Substitute (1, 550) and (3, 1100) into the slope formula to calculate the rate of change. Slope formula Substitute (1, 550) and (3, 1100) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. = 275 10 3.3.3: Recognizing Average Rate of Change

11 Guided Practice: Example 2, continued The rate of change for this function over the interval [1, 3] is approximately $275 per year. 11 3.3.3: Recognizing Average Rate of Change ✔

12 12 3.3.3: Recognizing Average Rate of Change Guided Practice: Example 2, continued 12

13 Guided Practice Example 3 Jasper is curious about how the rate of change differs for the interval [3, 6]. Calculate the rate of change using the graph from Example 2. 13 3.3.3: Recognizing Average Rate of Change

14 Guided Practice: Example 3, continued 1.Determine the interval to be observed. The interval to observe is [3, 6], or where 3 ≤ x ≤ 6. 14 3.3.3: Recognizing Average Rate of Change

15 Guided Practice: Example 3, continued 2.Identify the starting point of the interval. The x-value of the starting point is 3. The corresponding y-value is approximately 1,100. The starting point of the interval is (3, 1100). 15 3.3.3: Recognizing Average Rate of Change

16 Guided Practice: Example 3, continued 3.Identify the ending point of the interval. The x-value for the ending point is 6. The corresponding y-value is approximately 3,100. The ending point of the interval is (6, 3100). 16 3.3.3: Recognizing Average Rate of Change

17 Guided Practice: Example 3, continued 17 3.3.3: Recognizing Average Rate of Change

18 Guided Practice: Example 3, continued 4.Substitute (3, 1100) and (6, 3100) into the slope formula to calculate the rate of change. Slope formula Substitute (3, 1100) and (6, 3100) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. 18 3.3.3: Recognizing Average Rate of Change

19 Guided Practice: Example 3, continued The rate of change for this function over the interval [3, 6] is approximately $666.67 per year. Notice that the rate of change for the interval [3, 6] is much steeper than that of the interval [1, 3]. 19 3.3.3: Recognizing Average Rate of Change ✔

20 20 3.3.3: Recognizing Average Rate of Change Guided Practice: Example 3, continued

Download ppt "Introduction Previously, we calculated the rate of change of linear functions as well as intervals of exponential functions. We observed that the rate."

Similar presentations

Proving Average Rate of Change

Proving Average Rate of Change
Identifying the Average Rate of Change

Identifying the Average Rate of Change
Introduction Geometric sequences are exponential functions that have a domain of consecutive positive integers. Geometric sequences can be represented.

Introduction Geometric sequences are exponential functions that have a domain of consecutive positive integers. Geometric sequences can be represented.
Introduction Exponential functions are functions that can be written in the form f(x) = ab x, where a is the initial value, b is the rate of decay or growth,

Introduction Exponential functions are functions that can be written in the form f(x) = ab x, where a is the initial value, b is the rate of decay or growth,
Introduction Exponential equations in two variables are similar to linear equations in two variables in that there is an infinite number of solutions.

Introduction Exponential equations in two variables are similar to linear equations in two variables in that there is an infinite number of solutions.
Introduction The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can.

Introduction The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can.
Introduction When linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context.

Introduction When linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context.
Exponential Functions

Exponential Functions
Introduction The slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important.

Introduction The slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important.
Introduction A sector is the portion of a circle bounded by two radii and their intercepted arc. Previously, we thought of arc length as a fraction of.

Introduction A sector is the portion of a circle bounded by two radii and their intercepted arc. Previously, we thought of arc length as a fraction of.
Midpoints of line segments. Key concepts  Line continue infinitely in both directions, their length cannot be measured.  A Line Segment is a part of.

Midpoints of line segments. Key concepts  Line continue infinitely in both directions, their length cannot be measured.  A Line Segment is a part of.
Introduction The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can.

Introduction The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can.
Introduction The distance formula can be used to find solutions to many real-world problems. In the previous lesson, the distance formula was used to find.

Introduction The distance formula can be used to find solutions to many real-world problems. In the previous lesson, the distance formula was used to find.
Lesson 3.4 Constant Rate of Change (linear functions)

Lesson 3.4 Constant Rate of Change (linear functions)
Introduction In previous lessons, we have found the slope of linear equations and functions using the slope formula,. We have also identified the slope.

Introduction In previous lessons, we have found the slope of linear equations and functions using the slope formula,. We have also identified the slope.
Introduction In an equation with one variable, x, the solution will be the value that makes the equation true. For example: 1 is the solution for the equation.

Introduction In an equation with one variable, x, the solution will be the value that makes the equation true. For example: 1 is the solution for the equation.
Introduction In an equation with one variable, x, the solution will be the value that makes the equation true. For example: 1 is the solution for the equation.

Introduction In an equation with one variable, x, the solution will be the value that makes the equation true. For example: 1 is the solution for the equation.
Introduction Remember that linear functions are functions that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Introduction Remember that linear functions are functions that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.
Introduction Previously, we learned how to solve quadratic-linear systems by graphing and identifying points of intersection. In this lesson, we will focus.

Introduction Previously, we learned how to solve quadratic-linear systems by graphing and identifying points of intersection. In this lesson, we will focus.
Introduction We have worked with linear equations and equations of circles. We have solved systems of equations, including systems involving linear equations.

Introduction We have worked with linear equations and equations of circles. We have solved systems of equations, including systems involving linear equations.

Similar presentations

Ads by Google