Ch. 6 Linear Functions Notes

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1 First Name: Last Name: Block: Ch. 6 Linear Functions Notes 6.1 SLOPE OF A LINE Ch. 6.1 HW: p. 9 #4 1, 17,,, 8 6. SLOPES OF PARALLEL AND PERPENDICULAR LINES 6 Ch. 6. HW: p. 49 # 6 odd letters, INVESTIGATING GRAPHS OF LINEAR FUNCTIONS SLOPE INTERCEPT FORM OF THE EQUATION FOR A LINEAR FUNCTION 1 Ch. 6.4 HW: p. 6 # 4 9, 1, SLOPE POINT FORM OF THE EQUATION FOR A LINEAR FUNCTION 16 Ch. 6. HW: p. 7 #4 14, GENERAL FORM OF THE EQUATION FOR A LINEAR RELATION 0 Ch. 6.6 HW: p. 84 #4 9, 1 14, 18, 4 CH. 6 REVIEW LINEAR FUNCTIONS Created by Ms. Lee 1 of 6

2 6.1 Slope of a Line rise Definition: Slope = run In order to move from A to E, you need to move units and units. Rise = ; Run = rise The measure of of a line is called the slope. run Slope = In order to move from E to A, you need to move units and units. Rise = ; Run = rise The measure of of a line is called the slope. run Slope = Examples: Determining the Slope of a Line Segment 1) Find two exact points on the graph ) Determine the rise and run ) Determine the slope. Created by Ms. Lee of 6

3 Examples: Noticing the Patterns Examples: Drawing a Line Segment with a Given Slope and a Point. Draw a line segment with each given slope. 4 Slope = through (0, -) through (-4, ) Created by Ms. Lee of 6

4 Examples: Determining Slope Given Two Points on a Line. Rise y Slope = y Run x x 1 Determine the slope of the line that passes through C(-, ) and D(, 1) using the above formula. Determine the slope of the line that passes through C(-, ) and D(, 1) by sketching the line segment Determine the slope of the line that passes through A(-4, ) and B(, -) using the above formula. Determine the slope of the line that passes through A(-4, ) and B(, -) by sketching the line segment Created by Ms. Lee 4 of 6

5 Examples: Interpreting the Slope of a Line Yvonne recorded the distance she travelled at certain times since she began her cycling trip along the Trans Canada Trail in Manitoba, from North Winnipeg to Grand Beach. She plotted these data on a grid. a) What is the slope of the line through these points? b) What does the slope represent? c) How can the answer to part b be used to determine how far Yvonne travelled in 1 hours? 4 d) How can the answer to part b be used to determine the time it took Yvonne to travel km? Multiple Choice Question: 1) A line with an undefined slope passes through the points (-, 1) and (p, q). Which of the following points could be (p, q)? A. (1, 0) B. (0, 1) C. (0, -) D. (-, 0) Ch. 6.1 HW: p. 9 #4 1, 17,,, 8 Created by Ms. Lee of 6

Determine the slope of each line segment. Slope of Line A = Slope of Line B = If two lines are parallel, their slopes are.

6 6. Slopes of Parallel and Perpendicular Lines Definition: Parallel lines: Parallel lines are lines that never intersect (meet). Line A and Line B are parallel lines. Determine the slope of each line segment. Slope of Line A = Slope of Line B = Line A and Line B are parallel lines. Determine the slope of each line segment. Slope of Line A = Slope of Line B = If two lines are parallel, their slopes are. Examples: Identifying Parallel Lines Line AB passes through A(-, -) and B(-1, 6). Line CD passes through C(-1, -) and D(1, 7). Line EF passes through E(, -) and F(4, ). Sketch the lines. Are they parallel? Justify the answer. Created by Ms. Lee 6 of 6

Definition: Perpendicular lines: Perpendicular lines are lines that intersect at a 90 angle. Line A and Line B are perpendicular lines. Determine the slope of each line segment.

7 Definition: Perpendicular lines: Perpendicular lines are lines that intersect at a 90 angle. Line A and Line B are perpendicular lines. Determine the slope of each line segment. Slope of Line A = Slope of Line B = Line A and Line B are perpendicular lines. Determine the slope of each line segment. Slope of Line A = Slope of Line B = If two lines are perpendicular, their slopes are. Examples: Identifying a Line Perpendicular to a Given Line. a) Determine the slope of a line that is perpendicular to the line through E(,) and F(-4, -1). b) Determine the coordinates of G so that line EG is perpendicular to line EF. Created by Ms. Lee 7 of 6

8 Definitions: x- intercept: x-value where the line intercepts the x-axis. y-intercept: y-value where the line intercepts the y-axis. Examples: 1) AB has an x-intercept of and a y-intercept of -. CD has an x-intercept of and a y-intercept of. How are the lines related? Justify your answer. ) The coordinates of the vertices of ABC are A(-1, ), B(1, -) and C(4, ). Is it a right triangle? Justify your answer. Ch. 6. HW: p. 49 # 6 odd letters, 7 0 Created by Ms. Lee 8 of 6

9 6. Investigating Graphs of Linear Functions Using a table of values, graph the linear functions. Any pattern you notice? y = x x y Slope = y-intercept = y = 4x + 1 x y Slope = y-intercept = y = x x y Slope = y-intercept = y = x 4 x y Slope = y-intercept = Created by Ms. Lee 9 of 6

10 y = -x x y Slope = y-intercept = y = -x + x y Slope = y-intercept = y = -4x x y Slope = y-intercept = y = -x x y Slope = y-intercept = In general, given a linear function in the Slope-Intercept Form: y = mx + b, m is the and b is the Created by Ms. Lee 10 of 6

11 Questions: 1) What is the slope and y-intercept of each linear function? Linear functions Slope y-intercept a) y = x 4 1 b) y = x c) y = x d) y = x e) y = x 1 f) y = x g) y = x 4 h) y = x i) y = 4 j) y = - ) Which of the function(s) in Question 1 is parallel to y = x 7? 1 ) Which of the function(s) in Question 1 is parallel to y = x 1? 4) Which of the function(s) in Question 1 is perpendicular to y = x 1? ) Which of the function(s) in Question 1 is perpendicular to y = x? Created by Ms. Lee 11 of 6

12 Graph each linear function without using table of values. 1) y x 1 ) y x 1 ) 1 1 y x 4) y x ) 4 y x 1 6) y x Created by Ms. Lee 1 of 6

13 6.4 Slope-Intercept Form of the Equation for a Linear Function Recap: Given a linear function in the Slope-Intercept Form, y = mx + b, m = slope b = y-intercept Questions: 6) What is the slope and y-intercept of each linear function? k) y = x 4 l) y = x 7 m) f(x) = x 4 Examples: Writing an Equation of a Linear Function Given Its Slope and y-intercept 1) The graph of a linear function has slope and y-intercept. Write an equation for this function. 1 ) The graph of a linear function has slope and y-intercept. Write an equation for this function. Examples: Graphing a Linear Function Given its Equation in Slope-Intercept Form Graph each linear function: 1 1) y = x + ) y = x 4 Created by Ms. Lee 1 of 6

14 ) y = 4x 1 4) y = x 4 1 ) y = x 6 6) y = x Examples: Writing the equation of a Linear Function Given Its Graph Write an equation to describe this function. Created by Ms. Lee 14 of 6

15 Examples: Using an Equation of a Linear Function to Solve a Problem 1) The student council sponsored a dance. A ticket costs $ and the cost for the DJ was $00. a) Write an equation for the profit, P dollar, on the sale of t number tickets. b) Suppose 1 people bought tickets. What was the profit? c) Suppose the profit was $0. How many people bought tickets? d) Could the profit be exactly $146? Justify the answer. ) For a service call, an electrician charges a $70 initial fee, plus $60 for each hour she works. a) Write an equation to represent the total cost, C dollars, for t hours of work. b) What is the total cost for 4 hours of work? c) How would the equation change if the electrician charges $90 initial fee plus $0 for each hour she works? Ch. 6.4 HW: p. 6 # 4 9, 1, 1 16 Created by Ms. Lee 1 of 6

6. Slope-Point Form of the Equation for a Linear Function We can write an equation of a linear function in many different forms. So far, you know how to write the equation in Slope-Intercept Form.

Slope-Intercept Form: m slope b y int ercept y mx b This form is useful given the and Slope-Point Form: y y1 m( x x1 ) m slope x1 is the x-value of a point on the line y is the corresponding y-value

16 6. Slope-Point Form of the Equation for a Linear Function We can write an equation of a linear function in many different forms. So far, you know how to write the equation in Slope-Intercept Form. Today, we ll focus on Slope-Point Form. Slope-Intercept Form: m slope b y int ercept y mx b This form is useful given the and Slope-Point Form: y y1 m( x x1 ) m slope x1 is the x-value of a point on the line y is the corresponding y-value of the point 1 This form is useful given the and Examples: Writing an Equation (in Slope-Point Form) using a Point on the Line and Its Slope 1) Investigate by following the instructions below: Step1: Determine the slope, m, of the line segment Step : Write an equation in Slope-Point Form using the point A(-, -). Step : Write an equation in Slope-Point Form using the point B(1, 4). Step : Write the equation in step in Slope- Intercept Form (isolate y). Step : Write the equation in step in Slope- Intercept Form (isolate y). Created by Ms. Lee 16 of 6

17 ) A line goes through A(, ) and has the slope of -. a) Graph the line. b) Write the equation of the line in Slope-Point Form. c) Write the equation of the line in Slope- Intercept Form. ) A line goes through A(-1, 0) and has the slope of. d) Graph the line. e) Write the equation of the line in Slope-Point Form. f) Write the equation of the line in Slope- Intercept Form. Examples: Graphing a Linear Function Given Its Equation in Slope-Point Form 1) Given y ( x 1) a) Which point does the linear function c) Graph the line. go through? b) What is the slope? Created by Ms. Lee 17 of 6

18 ) Given y ( x ) a) Which point does the linear function go through? c) Graph the line. b) What is the slope? Examples: Writing an Equation of a Linear Function Given Two Points. 1) Write an equation for the line that passes through A(-, -) and C(1, -) First find the slope: Choose one point (it doesn t matter which one): a) Write the equation in slope-point form b) Write the equation in slope-intercept form (rearrange the above equation to isolate y). Created by Ms. Lee 18 of 6

19 ) Write an equation for the line that has a y-intercept of 4 and x-intercept of -. First find the slope: Choose one point (it doesn t matter which one): a) In slope-point form b) In slope-intercept form Examples: Writing an Equation of a Linear Function Given Two Points. 1) Write an equation for the line that passes through D(1, -1) and is: a) Parallel to the line y = x b) Perpendicular to the line y = x Ch. 6. HW: p. 7 #4 14, 19 Created by Ms. Lee 19 of 6

6.6 General Form of the Equation for a Linear Relation Different Forms of Linear Equation: Forms Equations Benefits Slope-Intercept Form y mx b It is easy to determine the slope, m, and y-intercept,

20 6.6 General Form of the Equation for a Linear Relation Different Forms of Linear Equation: Forms Equations Benefits Slope-Intercept Form y mx b It is easy to determine the slope, m, and y-intercept, b Slope-Point Form y y1 m( x x1 ) It is easy to determine the slope, m, and point, ( 1, y 1 General Form Ax By C 0 It is easy to determine the x int ercept and y int ercept of a line. A - Whole Number B & C - Integers Standard Form Ax + By = C A - Whole Number B & C - Integers It is easier to determine the x int ercept and y int ercept of a line. Examples: Graphing a Line in General Form 1) Graph the line defined by x y 18 0 Step1: Determine x-intercept Step : Plot x- and y-intercepts and graph the line Step: Determine y-intercept Created by Ms. Lee 0 of 6

21 ) Graph the line defined by x y 9 0 Step1: Determine x-intercept Step : Plot x- and y-intercepts and graph the line Step: Determine y-intercept Examples: Determining the Slope of a Line Given Its Equation in General Form ) Determine the slope of the line with this equation: x y ) Determine the slope of the line defined by: x y 1 0 Created by Ms. Lee 1 of 6

22 Examples: Rewriting an Equation in General Form ) Write each equation in general form. a) 1 y x 4 c) y ( x 4) b) y 1 ( x ) d) y x 4. Ch. 6.6 HW: p. 84 #4 9, 1 14, 18, 4 Created by Ms. Lee of 6

23 Ch. 6 Review Linear Functions Multiple Choice: 1. Determine the slope of a line perpendicular to y x 1. a) b) c) d). Determine the slope of a line parallel to y 4x. a) 4 b) c) d) 4 4. Determine the slope and y-intercept of the line, 1 y x a) slope =, y-int = 4 b) slope =, y-int = -4 c) slope = -, y-int = 4 d) slope =, y-int = Which of the following points lies on the graph of y ( x 4) a) (-4, -) b) (4, -) c) (-4, ) d) (4, -). Which of the following equations describes the linear relation graphed below? I. 4 y x 4 II. 4 y 8 ( x ) III. 4x y 1 0 a) I only b) II and III only c) I and II only d) I and III only Created by Ms. Lee of 6

6. Which graph represents the relation: x y 10 0? a) b) c) d) 7.

passes through the points (6, 1) and (-10, 9).

A line with an undefined slope passes through the points (-, 1) and

24 6. Which graph represents the relation: x y 10 0? a) b) c) d) 7. Calculate the slope between the points (7, -) and (4, ). 1 b) a) c) d) Determine the equation of a line, in slope-intercept form, that passes through the points (6, 1) and (-10, 9). 1 1 a) y x 4 b) y x c) y x 8 d) y x 1 9. A line with an undefined slope passes through the points (-, 1) and (p, q). Which of the following points could be (p, q)? a) (1, 0) b) (0, 1) c) (0, -) d) (-, 0) 10. Determine the slope of the linear relation x y a) b) c) d) Created by Ms. Lee 4 of 6

25 11. Determine the slope of the linear relation x y 1 0 a) c) b) d) 1. Which of the following coordinates are intercepts of the linear relation x y 0 0 I. (0, 10) II. ( 0, ) III. ( 10,0) IV. ( 1,0) a) I only b) I and IV only c) II and III only d) II and IV only 1. Determine the slope-intercept equation of the line that is parallel to y x and passes through the point (0, ) a) y x b) y x c) y x d) y x Written Response: 14. Determine the equation of a line that passes through A(-6, 8) and C(-1, -) [ marks] a) in slope-point form: y y m x ) 1 ( x1 b) in slope-intercept form: y mx b. Created by Ms. Lee of 6

For each equation, draw the graph. [ marks] a) y x 1 b) y 1 ( x ) 4 17.

26 1. Determine the equation of a line through A(, -4) that is perpendicular to y x 1. Write the equation [ marks] a) In slope-point form: y y m x ) 1 ( x1 b) In general form: Ax By C For each equation, draw the graph. [ marks] a) y x 1 b) y 1 ( x ) Write an equation for the graph in slope-point form. [1 mark] 18. Write an equation for the graph in slope-intercept form. [1 mark] Created by Ms. Lee 6 of 6

6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line.

6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line. Toolkit: - Rate of change - Simplifying fractions Main Ideas: Definitions Rise: the vertical distance between two