Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions
First Name: ______Last Name: ______Block: ______
Ch. 6 – Linear Functions Notes
Ch. 6.1 HW: p. 339 #4 – 13, 17, 23, 25, 28 5
6. 2 – SLOPES OF PARALLEL AND PERPENDICULAR LINES 6
Ch. 6.2 HW: p. 349 #3 – 6 odd letters, 7 – 20 8
6.3 – INVESTIGATING GRAPHS OF LINEAR FUNCTIONS 9
6.4 – SLOPE‐INTERCEPT FORM OF THE EQUATION FOR A LINEAR FUNCTION 13
Ch. 6.4 HW: p. 362 # 4 – 9, 12, 13 – 16 15
6.5 – SLOPE‐POINT FORM OF THE EQUATION FOR A LINEAR FUNCTION 16
Ch. 6.5 HW: p. 372 #4 – 14, 19 – 25 19
6.6 – GENERAL FORM OF THE EQUATION FOR A LINEAR RELATION 20
Ch. 6.6 HW: p. 384 #4 – 9, 12 – 14, 18, 22 – 24 22
CH. 6 REVIEW – LINEAR FUNCTIONS 23
Created by Ms. Lee 1 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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 2) Determine the rise and run 3) Determine the slope.
Created by Ms. Lee 2 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions
Examples: Noticing the Patterns
Examples: Drawing a Line Segment with a Given Slope and a Point. Draw a line segment with each given slope. 2 4 Slope = through (0, -2) through (-4, 3) 3 5
Created by Ms. Lee 3 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions Examples: Determining Slope Given Two Points on a Line. Rise y y Slope = 2 2 Run x2 x1
Determine the slope of the line that passes Determine the slope of the line that passes through C(-3, 5) and D(2, 1) using the above through C(-3, 5) and D(2, 1) by sketching the formula. line segment
Determine the slope of the line that passes Determine the slope of the line that passes through A(-4, 3) and B(2, -5) using the above through A(-4, 3) and B(2, -5) by sketching the formula. line segment
Created by Ms. Lee 4 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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?
3 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 55km?
Multiple Choice Question: 1) A line with an undefined slope passes through the points (-2 , 1) and (p , q). Which of the following points could be (p , q)? A. (1, 0) B. (0 , 1) C. (0 , -2) D. (-2 , 0) Ch. 6.1 HW: p. 339 #4 – 13, 17, 23, 25, 28
Created by Ms. Lee 5 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 6. 2 – 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(-3, -2) and B(-1, 6). Line CD passes through C(-1, -3) and D(1, 7). Line EF passes through E(2, -5) and F(4, 3). Sketch the lines. Are they parallel? Justify the answer.
Created by Ms. Lee 6 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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(2,3) 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 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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 3 and a y-intercept of -2. CD has an x-intercept of 2 and a y-intercept of 3. How are the lines related? Justify your answer.
2) The coordinates of the vertices of ABC are A(-1, 3), B(1, -2) and C(4, 5). Is it a right triangle? Justify your answer.
Ch. 6.2 HW: p. 349 #3 – 6 odd letters, 7 – 20
Created by Ms. Lee 8 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 6.3 – Investigating Graphs of Linear Functions
Using a table of values, graph the linear functions. Any pattern you notice? y = 2x x y Slope =
y-intercept =
y = 4x + 1 x y Slope =
y-intercept =
y = x – 2 x y Slope =
y-intercept =
y = 3x – 4 x y Slope =
y-intercept =
Created by Ms. Lee 9 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions y = -3x x y Slope =
y-intercept =
y = -x + 2 x y Slope =
y-intercept =
y = -4x – 3 x y Slope =
y-intercept =
y = -2x – 5 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 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions Questions:
1) What is the slope and y-intercept of each linear function? Linear functions Slope y-intercept a) y = 3x 4
1 b) y = x 5 2
3 c) y = x 2
d) y = 2x 3
2 e) y = x 1 5
3 f) y = x 5
g) y = x 4
h) y = 3x 5
i) y = 4
j) y = -3
2) Which of the function(s) in Question 1 is parallel to y = 3x 7 ?
1 3) Which of the function(s) in Question 1 is parallel to y = x 1? 2
4) Which of the function(s) in Question 1 is perpendicular to y = x 1?
5 5) Which of the function(s) in Question 1 is perpendicular to y = x 2 ? 2
Created by Ms. Lee 11 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions Graph each linear function without using table of values. 1) y 3x 1 2) y 3x 1
1 1 3) y x 3 4) y x 3 2 2
4 2 5) y x 1 6) y x 3 3
Created by Ms. Lee 12 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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 = 3x 4
2 l) y = x 5 7
3 m) f(x) = x 2 4
Examples: Writing an Equation of a Linear Function Given Its Slope and y-intercept 1) The graph of a linear function has slope 3 and y-intercept 5. Write an equation for this function.
1 2) The graph of a linear function has slope and y-intercept 3 . Write an equation for this 2 function.
Examples: Graphing a Linear Function Given its Equation in Slope-Intercept Form Graph each linear function: 1 2 1) y = x + 5 2) y = x 4 2 3
Created by Ms. Lee 13 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions
3) y = 4x 1 4) y = 3x 2
4 1 5) y = x 6 6) y = x 2 5 3
Examples: Writing the equation of a Linear Function Given Its Graph Write an equation to describe this function.
Created by Ms. Lee 14 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions
Examples: Using an Equation of a Linear Function to Solve a Problem
1) The student council sponsored a dance. A ticket costs $5 and the cost for the DJ was $300.
a) Write an equation for the profit, P dollar, on the sale of t number tickets.
b) Suppose 123 people bought tickets. What was the profit?
c) Suppose the profit was $350. How many people bought tickets?
d) Could the profit be exactly $146? Justify the answer.
2) 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 $50 for each hour she works?
Ch. 6.4 HW: p. 362 # 4 – 9, 12, 13 – 16
Created by Ms. Lee 15 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 6.5 – 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: y mx b Slope-Point Form: y y1 m(x x1) m slope m slope b y int ercept x1is the x-value of a point on the line
y1 is the corresponding y-value of the point
This form is useful given the ______This form is useful given the ______and ______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 2: Write an equation in Slope-Point Form Step 2: Write an equation in Slope-Point Form using the point A(-2, -5). using the point B(1, 4).
Step 3: Write the equation in step 2 in Slope- Step 3: Write the equation in step 2 in Slope- Intercept Form (isolate y). Intercept Form (isolate y).
Created by Ms. Lee 16 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 2) A line goes through A(3, 5) and has the slope of -2. 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.
3) A line goes through A(-1, 0) and has the slope of 3. 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 2 3(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 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 2 2) Given y 5 (x 3) 3 a) Which point does the linear function c) Graph the line. go through?
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(-2, -5) and C(1, -2) 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 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 2) Write an equation for the line that has a y-intercept of 4 and x-intercept of -3. 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: 2 a) Parallel to the line y = x 3 5
2 b) Perpendicular to the line y = x 3 5
Ch. 6.5 HW: p. 372 #4 – 14, 19 – 25
Created by Ms. Lee 19 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 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, ( x1, y1 ) 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 It is easier to determine the x int ercept and y int ercept of a line. A - Whole Number B & C - Integers
Examples: Graphing a Line in General Form 1) Graph the line defined by 3x 2y 18 0 Step1: Determine x-intercept Step 3: Plot x- and y-intercepts and graph the line
Step2: Determine y-intercept
Created by Ms. Lee 20 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 2) Graph the line defined by x 3y 9 0 Step1: Determine x-intercept Step 3: Plot x- and y-intercepts and graph the line
Step2: Determine y-intercept
Examples: Determining the Slope of a Line Given Its Equation in General Form 3) Determine the slope of the line with this equation: 3x 2y 16 0
4) Determine the slope of the line defined by: 5x 2y 12 0
Created by Ms. Lee 21 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions Examples: Rewriting an Equation in General Form 5) Write each equation in general form. 1 3 a) y x 3 c) y 2 (x 4) 4 2
3 2 b) y 1 (x 2) d) y x 4 . 5 3
Ch. 6.6 HW: p. 384 #4 – 9, 12 – 14, 18, 22 – 24
Created by Ms. Lee 22 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions Ch. 6 Review – Linear Functions
Multiple Choice: 2 1. Determine the slope of a line perpendicular to y x 1. 3 3 3 a) b) 2 2 2 2 c) d) 3 3
2. Determine the slope of a line parallel to y 4x 5. a) 4 b) -4 1 1 c) d) 4 4
1 3. Determine the slope and y-intercept of the line, y x 4 . 3 1 1 a) slope = , y-int = 4 b) slope = , y-int = -4 3 3 c) slope = -3, y-int = 4 d) slope = 3, y-int = -4
4. Which of the following points lies on the graph of y 3 3(x 4) a) (-4, -3) b) (4, -3) c) (-4, 3) d) (4, -3)
5. Which of the following equations describes the linear relation graphed below?
4 I. y x 4 3 4 II. y 8 (x 3) 3
III. 4x 3y 12 0
a) I only b) II and III only c) I and II only d) I and III only
Created by Ms. Lee 23 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 6. Which graph represents the relation: x 5y 10 0 ? a) b)
c) d)
7. Calculate the slope between the points (7, -3) and (4, 3). 1 b) 2 a) 2 c) 2 d) 10
8. 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 2 2 2 c) y 2x 8 d) y 2x 13
9. A line with an undefined slope passes through the points (-2, 1) and (p, q). Which of the following points could be (p, q)? a) (1, 0) b) (0, 1) c) (0, -2) d) (-2, 0)
10. Determine the slope of the linear relation 5x 2y 2 5 5 a) b) 2 2 2 2 c) d) 5 5
Created by Ms. Lee 24 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 11. Determine the slope of the linear relation 3x 5y 15 0 5 3 a) b) 3 5 5 3 c) d) 3 5
12. Which of the following coordinates are intercepts of the linear relation 2x 3y 30 0 I. (0, 10) 2 II. (0, ) 3 III. (10,0) IV. (15,0) a) I only b) I and IV only c) II and III only d) II and IV only
2 13. Determine the slope-intercept equation of the line that is parallel to y x 3 and passes through 3 the point (0, 5) 3 3 a) y x 5 b) y x 5 2 2 2 2 c) y x 5 d) y x 5 3 3
Written Response: 14. Determine the equation of a line that passes through A(-6, 8) and C(-1, -2) [3 marks]
a) in slope-point form: y y1 m(x x1)
b) in slope-intercept form: y mx b .
Created by Ms. Lee 25 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson
Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions 2 15. Determine the equation of a line through A(2, -4) that is perpendicular to y x 1. Write the 3 equation [2 marks] a) In slope-point form: y y1 m(x x1)
b) In general form: Ax By C 0 .
16. For each equation, draw the graph. [2 marks] a) y 2x 3 1 b) y 1 (x 2) 4
17. 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 26 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson