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Introduction

Introduction

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Introduction

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  1. Introduction Tables and graphs can be represented by equations. Data represented in a table can either be analyzed as a pattern, like the data presented in the previous lesson, or it can be graphed first. The shape of a graph helps identify which type of equation can be used to represent the data. Data that is represented using a linear equation has a constant slope and the graph is a straight line. 3.6.2: Constructing Functions from Graphs and Tables

  2. Introduction, continued The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The y-interceptis the point at which the graph intersects the y-axis. To find the equation of a line, first find the slope, then replace x and y in the general equation with x and y from an ordered pair on the graph. Then, solve the equation for b. 3.6.2: Constructing Functions from Graphs and Tables

  3. Introduction, continued The slope of a line is the change in the independent variable divided by the change in the dependent variable, and describes the rate of change of the variables. Using the points (x1, y1) and (x2, y2), the slope can be calculated by finding . The slope is similar to the common difference between terms, except the slope can be found between points that have x-values that are more than one unit apart. 3.6.2: Constructing Functions from Graphs and Tables

  4. Introduction, continued The slope allows us to compare the change between any two points on a graph or in a table. An exponential equation has a slope that is constantly changing, either increasing or decreasing.Graphically, exponential equations are a curve. The general form of an exponential equation is y = ab x, where a and b are real numbers, and b is the common ratio. The values of a and b change the shape of the graph of an exponential function. On the following slides, you’ll find four examples of functions with different a and b values. 3.6.2: Constructing Functions from Graphs and Tables

  5. Introduction, continued a > 0 and b > 1 f (x) = 2 x 3.6.2: Constructing Functions from Graphs and Tables

  6. Introduction, continued a > 0 and 0 < b < 1 f (x) = (0.5) x 3.6.2: Constructing Functions from Graphs and Tables

  7. Introduction, continued a < 0 and b > 1 f (x) = (–1) • 2x 3.6.2: Constructing Functions from Graphs and Tables

  8. Introduction, continued a < 0 and 0 < b < 1 3.6.2: Constructing Functions from Graphs and Tables

  9. Introduction, continued Data that follows an exponential pattern has a common ratio between the dependent quantities. When looking for the common ratio, be sure to verify that the x-values, or independent quantities, are each one unit apart. The common ratio, b, can be used to write an equation to represent the exponential pattern. Find the value of the equation at x = 0,f (0). The general equation of the function is f (x) =f (0) • bx. 3.6.2: Constructing Functions from Graphs and Tables

  10. Key Concepts The graph of a linear equation is a straight line. Linear equations have a constant slope, or rate of change. Linear equations can be written as functions. The general form of a linear function is f (x) = mx + b, where m is the slope and b is the y-intercept. The slope of a linear function can be calculated using any two points, (x1, y1) and (x2, y2): the formula is . 3.6.2: Constructing Functions from Graphs and Tables

  11. Key Concepts, continued The y-intercept is the point at which the graph of the equation crosses the y-axis. Exponential equations have a slope that is constantly changing. Exponential equations can be written as functions. The general form of an exponential function is f(x) = abx, where a and b are real numbers. 3.6.2: Constructing Functions from Graphs and Tables

  12. Key Concepts, continued The graph of an exponential equation is a curve. The common ratio, b, between independent quantities in an exponential pattern, and the value of the equation at x = 0, f(0), can be used to write the general equation of the function: f(x) = f (0) • bx. 3.6.2: Constructing Functions from Graphs and Tables

  13. Common Errors/Misconceptions looking for a common difference between dependent quantities when the independent values are not one unit apart incorrectly calculating the slope trying to match an exponential graph to a linear equation trying to match a linear graph to an exponential equation 3.6.2: Constructing Functions from Graphs and Tables

  14. Guided Practice Example 2 Determine the equation that represents the relationship between xand y in the graph to the right. 3.6.2: Constructing Functions from Graphs and Tables

  15. Guided Practice: Example 2, continued Determine which type of equation, linear or exponential, will fit the graph. The graph of a linear equation is a straight line, and a graph of an exponential equation is a curve. A linear equation can be used to represent the graph. 3.6.2: Constructing Functions from Graphs and Tables

  16. Guided Practice: Example 2, continued Identify at least three points from the graph. From the graph, three points are: (0, –4), (1, –1), and (2, 2). 3.6.2: Constructing Functions from Graphs and Tables

  17. Guided Practice: Example 2, continued Find the slope of the line, using any two of the points. The slope of the line is for any two points (x1, y1) and (x2, y2). Using the points (0, –4) and (1, –1), the slope is . 3.6.2: Constructing Functions from Graphs and Tables

  18. Guided Practice: Example 2, continued Find the y-intercept of the line. The y-intercept can either be found by solving the equation f(x) = mx + b for b, or by finding the value of y when x = 0. On the graph, we can see the point (0, –4). The y-intercept is –4. 3.6.2: Constructing Functions from Graphs and Tables

  19. Guided Practice: Example 2, continued Use the slope and y-intercept to find an equation of the line. The general form of the linear function is f (x) = mx + b, where m is the slope and b is the y-intercept. The equation to represent the line is f (x) = 3x – 4. Check the equation by evaluating it at the values of x from previously identified points. x = 0, f (0) = 3(0) – 4 = –4 x = 1, f (1) = 3(1) – 4 = –1 x = 2, f (2) = 3(2) – 4 = 2 The relationship can be represented using the equation f (x) = 3x – 4. ✔ 3.6.2: Constructing Functions from Graphs and Tables

  20. Guided Practice: Example 2, continued 20 3.6.2: Constructing Functions from Graphs and Tables

  21. Guided Practice Example 3 A clothing store discounts items on a regular schedule. Each week, the price of an item is reduced. The prices for one item are in the table to the right. Week 0 is the starting price of the item. Determine a linear or exponential equation that represents the relationship between the week and the price of the item. 3.6.2: Constructing Functions from Graphs and Tables

  22. Guided Practice: Example 3, continued Create a graph of the data. Let the x-axis represent the week, and the y-axis represent the price in dollars. 3.6.2: Constructing Functions from Graphs and Tables

  23. Guided Practice: Example 3, continued Determine if a linear or exponential equation could represent the data. The x-values of the points vary by 1. Look at the vertical distance between each pair of points. It appears to be decreasing, and is not remaining constant. An exponential equation could represent the data. 3.6.2: Constructing Functions from Graphs and Tables

  24. Guided Practice: Example 3, continued Find the common ratio between the terms. The four points are at the x-values 0, 1, 2, and 3, so the x-values are each one unit apart. Look at the pattern of the y-values: 100, 60, 36, and 21.60. Divide each y-value by the previous y-value to identify the common ratio. 3.6.2: Constructing Functions from Graphs and Tables

  25. Guided Practice: Example 3, continued The common ratio is 0.60. 3.6.2: Constructing Functions from Graphs and Tables

  26. Guided Practice: Example 3, continued Use the value of the equation at x = 0 and the common ratio to write an equation to represent the relationship. At week 0, the price is $100. The common ratio is 0.60. An equation to represent the relationship is f(x) = 100 • (0.60)x. Evaluate the equation at the given values of x to check the equation. x = 0, f (0) = 100 • (0.60)0 = 100 x = 1, f (1) = 100 • (0.60)1 = 60 x = 2, f (2) = 100 • (0.60)2 = 36 x = 3, f (3) = 100 • (0.60)3 = 21.6 3.6.2: Constructing Functions from Graphs and Tables

  27. The price of the clothing item, y, at any week, x, can be represented by the equation f (x) = 100 • (0.60) x. ✔ 3.6.2: Constructing Functions from Graphs and Tables

  28. Guided Practice: Example 3, continued 3.6.2: Constructing Functions from Graphs and Tables

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