Chapter 3: Straight Lines and Linear Functions

Transcription

1 Chapter 3: Straight Lines and Linear Functions E. Smith MAT 1020 E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

2 Section 3.1 Key Ideas The definition of a linear function The meaning of the slope of a line The relationship between slope and ARC Solving geometric problems involving straight lines via slope Horizontal and Vertical Intercepts E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

3 Recall that the formula for the average rate of change is f (b) f (a) b a E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

4 Recall that the formula for the average rate of change is f (b) f (a) b a Example 1: Suppose f (x) measures the population of a protected species (in number of animals) t years after that species has been introduced to a wildlife preserve. What would the average rate of change of f (x) from x = 5 to x = 10 tell you in terms of the population? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

5 Linear Function 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

6 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

7 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 Some characteristics of a linear function: 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

8 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 Some characteristics of a linear function: Can be written in the form y = mx + b. 2 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

9 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 Some characteristics of a linear function: Can be written in the form y = mx + b. 2 The slope of a linear function is m = f (b) f (a) b a = ARC 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

10 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 Some characteristics of a linear function: Can be written in the form y = mx + b. 2 f (b) f (a) The slope of a linear function is m = b a = ARC The slope, or rate of change, m, of a line shows how steeply it is increasing or decreasing. It tells the vertical change along the line when there is a horizontal change of 1 unit. 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

11 Linear Function: Any function who s average rate of change between any two points is always equal and when graphed makes a straight line. 1 Some characteristics of a linear function: Can be written in the form y = mx + b. 2 f (b) f (a) The slope of a linear function is m = b a = ARC The slope, or rate of change, m, of a line shows how steeply it is increasing or decreasing. It tells the vertical change along the line when there is a horizontal change of 1 unit. If m = 0, the line is horizontal. 1 Although we call these linear functions, they are actually one variable affine functions. 2 This is often called Y-Intercept Form, where b is the y-intercept. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

12 Example 2: 1 If a linear function has a positive slope what do we know about the graph of the function? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

13 Example 2: 1 If a linear function has a positive slope what do we know about the graph of the function? 2 If a linear function has a negative slope what do we know about the graph of the function? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

14 Example 2: 1 If a linear function has a positive slope what do we know about the graph of the function? 2 If a linear function has a negative slope what do we know about the graph of the function? 3 If f (x) has a larger slope in absolute value than g(x) what do we know about how the shapes of the graphs of f (x) and g(x) compare? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

15 We can also use lines and their properties to solve problems. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

16 We can also use lines and their properties to solve problems. Example 3: Three horizontal feet north of a 10-foot tall maple tree is a 4-foot tall forsythia. We wish to place a spotlight north of the forsythia so that the light just hits the top of both the forsythia and the maple tree. How many feet north of the forsythia should we place the light. Note: We wish for the spotlight to be on the ground pointing upwards. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

17 Example 4: Plywood siding is to be used to cover the exterior wall of a house. Plywood siding comes in sheets 4 feet wide and 8 feet high. The piece on the far right is 1 foot high on the shorter side and 2 feet 6 inches high on the longer side (towards the peak of the roof). The final cut for the plywood would be the length k. What is the length? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

18 Section 3.2: Linear Functions Section 3.2 Key Ideas Understanding the meaning of slope, vertical intercept, and horizontal intercept and begging able to give practical information of them. Finding equations of lines given the slope (or enough information to find the slope) and a point. Finding equations of lines given two points. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

19 Section 3.2: Linear Functions Review E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

20 Section 3.2: Linear Functions Review What makes a function linear? E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

21 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

22 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

23 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. The slope of a linear function is give by m = f (b) f (a) b a = y 2 y 1 x 2 x 1 E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

24 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. The slope of a linear function is give by m = f (b) f (a) b a = y 2 y 1 x 2 x 1 Positive slope mean: increasing from left to right (uphill). E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

25 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. The slope of a linear function is give by m = f (b) f (a) b a = y 2 y 1 x 2 x 1 Positive slope mean: increasing from left to right (uphill). Negative slope means: decreasing from left to right (downhill). E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

26 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. The slope of a linear function is give by m = f (b) f (a) b a = y 2 y 1 x 2 x 1 Positive slope mean: increasing from left to right (uphill). Negative slope means: decreasing from left to right (downhill). X-Intercepts: When the y-value is equal to zero. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

27 Section 3.2: Linear Functions Review What makes a function linear? Its slope is constant (always the same). Its graph is a straight line. The slope of a linear function is give by m = f (b) f (a) b a = y 2 y 1 x 2 x 1 Positive slope mean: increasing from left to right (uphill). Negative slope means: decreasing from left to right (downhill). X-Intercepts: When the y-value is equal to zero. Y-Intercepts: When the x-value is equal to zero. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10

28 Section 3.2: Linear Functions Example 1: The value of a car in dollars t years after it is purchased is given by V (t) = t. What is the slope of V (t) and what does it mean in practical terms? What is the vertical intercept of V (t) and what does it mean in practical terms? Show that t = 11 is the horizontal intercept of V (t) and explain what this means in practical terms. E. Smith Chapter 3: Straight Lines and Linear Functions MAT / 10