Ch. 6 – Linear Functions Notes · 2017-06-19 · Foundations of Mathematics & Pre-Calculus 10...

Foundations of Mathematics & Pre-Calculus 10 Chapter 6 – Linear Functions

Created by Ms. Lee 1 of 26 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson

First Name: ________________________ Last Name: ________________________ Block: ______

Ch. 6 – Linear Functions Notes

6.1 – SLOPE OF A LINE 2

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



6.1–SlopeofaLine

Definition: Slope = run

rise

In order to move from A to E, you need to move ____ units _____ and ____ units _______. Rise = ________ ; Run = ________

The measure of run

rise of a line is called the slope.

Slope =

In order to move from E to A, you need to move ____ units _____ and ____ units _______. Rise = ________ ; Run = ________

The measure of run

rise of a line is called the slope.

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.



Examples: Noticing the Patterns

Examples: Drawing a Line Segment with a Given Slope and a Point. Draw a line segment with each given slope.

Slope = 3

2 through (0, -2)

5

4 through (-4, 3)



Examples: Determining Slope Given Two Points on a Line.

Slope = Run

Rise

12

22

xx

yy

Determine the slope of the line that passes through C(-3, 5) and D(2, 1) using the above formula.

Determine the slope of the line that passes through C(-3, 5) and D(2, 1) by sketching the line segment

Determine the slope of the line that passes through A(-4, 3) and B(2, -5) using the above formula.

Determine the slope of the line that passes through A(-4, 3) and B(2, -5) by sketching the line segment



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 4

31 hours?

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



6.2–SlopesofParallelandPerpendicularLines 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.



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.



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



6.3–InvestigatingGraphsofLinearFunctions 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


y = x – 2 x y


y = 3x – 4 x y




y = -3x x y


y = -x + 2 x y


y = -4x – 3 x y


y = -2x – 5 x y


In general, given a linear function in the Slope-Intercept Form: y = mx + b, m is the _____________ and b is the ___________________



Questions: 1) What is the slope and y-intercept of each linear function?

Linear functions Slope y-intercept a) y = 43 x

b) y = 52

1x

c) y = x2

3

d) y = 32 x

e) y = 15

2x

f) y = 5

3x

g) y = 4x

h) y = 53 x

i) y = 4

j) y = -3

2) Which of the function(s) in Question 1 is parallel to y = 73 x ?

3) Which of the function(s) in Question 1 is parallel to y = 12

1x ?

4) Which of the function(s) in Question 1 is perpendicular to y = 1 x ?

5) Which of the function(s) in Question 1 is perpendicular to y = 22

5 x ?



Graph each linear function without using table of values. 1) 13 xy

2) 13 xy

3) 32

1 xy

4) 32

1 xy

5) 13

4 xy

6) xy3

2



6.4–Slope-InterceptFormoftheEquationforaLinearFunction 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 = 43 x

l) y = 57

2x

m) f(x) = 24

3 x

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.

2) The graph of a linear function has slope 2

1 and y-intercept 3 . Write an equation for this

function.

Examples: Graphing a Linear Function Given its Equation in Slope-Intercept Form Graph each linear function:

1) y = 2

1x + 5

2) y = 43

2x



3) y = 14 x

4) y = 23 x

5) y = 65

4x

6) y = 23

1 x

Examples: Writing the equation of a Linear Function Given Its Graph Write an equation to describe this function.



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



6.5–Slope-PointFormoftheEquationforaLinearFunction 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: bmxy

slopem erceptyb int

This form is useful given the ____________ and _____________________

Slope-Point Form: )( 11 xxmyy slopem

1x is the x-value of a point on the line

1y is the corresponding y-value of the point 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 2: Write an equation in Slope-Point Form

using the point A(-2, -5).

Step 3: Write the equation in step 2 in Slope-

Intercept Form (isolate y).

Step 2: Write an equation in Slope-Point Form using the point B(1, 4).

Step 3: Write the equation in step 2 in Slope-

Intercept Form (isolate y).



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 )1(32 xy

a) Which point does the linear function go through?

b) What is the slope?

c) Graph the line.



2) Given )3(3

25 xy

a) Which point does the linear function go through?

b) What is the slope?

c) Graph the line.

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).



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:

a) Parallel to the line y = 35

2x

b) Perpendicular to the line y = 35

2x

Ch. 6.5 HW: p. 372 #4 – 14, 19 – 25



6.6–GeneralFormoftheEquationforaLinearRelation Different Forms of Linear Equation: Forms Equations Benefits Slope-Intercept Form bmxy It is easy to determine the slope, m , and y-intercept,

b Slope-Point Form )( 11 xxmyy It is easy to determine the slope, m , and point, (

11, yx ) General Form 0 CByAx

A - Whole Number B & C - Integers

It is easy to determine the erceptx int and ercepty int of a line.

Standard Form Ax + By = C A - Whole Number B & C - Integers

It is easier to determine the erceptx int and ercepty int of a line.

Examples: Graphing a Line in General Form 1) Graph the line defined by 01823 yx

Step1: Determine x-intercept Step2: Determine y-intercept

Step 3: Plot x- and y-intercepts and graph the line



2) Graph the line defined by 093 yx Step1: Determine x-intercept Step2: Determine y-intercept

Step 3: Plot x- and y-intercepts and graph the line

Examples: Determining the Slope of a Line Given Its Equation in General Form 3) Determine the slope of the line with this equation: 01623 yx

4) Determine the slope of the line defined by: 01225 yx



Examples: Rewriting an Equation in General Form 5) Write each equation in general form.

a) 34

1 xy

b) )2(5

31 xy

c) )4(2

32 xy

d) 43

2 xy .

Ch. 6.6 HW: p. 384 #4 – 9, 12 – 14, 18, 22 – 24



Ch.6Review–LinearFunctions Multiple Choice:

1. Determine the slope of a line perpendicular to 13

2 xy .

a) 2

3 b)

2

3

c) 3

2 d)

3

2

2. Determine the slope of a line parallel to 54 xy .

a) 4 b) -4

c) 4

1 d)

4

1

3. Determine the slope and y-intercept of the line, 43

1 xy .

a) slope = 3

1, y-int = 4 b) slope =

3

1, y-int = -4

c) slope = -3, y-int = 4 d) slope = 3, y-int = -4 4. Which of the following points lies on the graph of )4(33 xy

a) (-4, -3) b) (4, -3) c) (-4, 3) d) (4, -3)

5. Which of the following equations describes the linear relation graphed below?

I. 43

4 xy

II. )3(3

48 xy

III. 01234 yx

a) I only b) II and III only c) I and II only d) I and III only



6. Which graph represents the relation: 0105 yx ? a)

b)

c)

d)

7. Calculate the slope between the points (7, -3) and (4, 3).

a) 2

1

b) 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).

a) 42

1 xy b) 2

2

1 xy

c) 82 xy d) 132 xy 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 225 yx

a) 2

5 b)

2

5

c) 5

2 d)

5

2



11. Determine the slope of the linear relation 01553 yx

a) 3

5 b)

5

3

c) 3

5 d)

5

3

12. Which of the following coordinates are intercepts of the linear relation 03032 yx

I. (0, 10)

II. )3

2,0(

III. )0,10( IV. )0,15( a) I only b) I and IV only c) II and III only d) II and IV only

13. Determine the slope-intercept equation of the line that is parallel to 33

2 xy and passes through

the point (0, 5)

a) 52

3 xy b) 5

2

3 xy

c) 53

2 xy d) 5

3

2 xy

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: )( 11 xxmyy

b) in slope-intercept form: bmxy .



15. Determine the equation of a line through A(2, -4) that is perpendicular to 13

2 xy . Write the

equation [2 marks] a) In slope-point form: )( 11 xxmyy

b) In general form: 0 CByAx .

16. For each equation, draw the graph. [2 marks]

a) 32 xy

b) )2(4

11 xy

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]

Ch. 6 – Linear Functions Notes · 2017-06-19 · Foundations of Mathematics & Pre-Calculus 10...

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Transcript of Ch. 6 – Linear Functions Notes · 2017-06-19 · Foundations of Mathematics & Pre-Calculus 10...