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Math 10C

Linear Functions

STUDENT NAME: ________________________________

CHAPTER EXAM DATE: _________________________

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Math 10C

Unit: Linear Functions

Topic: Slope of a line

Objectives:

• Determine the slope of a line segment and of a line.

NOTES

Slope of a line segment is the measure of its rate of change. Slope can be written many ways.

A B

A B

y yrise ym

run x x x

−= = =

−

When a line segment goes up from left to right both x and y increase so the slope is positive. When a

line segment goes down from left to right y decreases as x increases so the slope of the line is negative.

A horizontal line has no change in y as x increases so the slope is zero. A vertical line has an

increasing y but no change in x so the slope is undefined.

All line segments of a line have equal slope. So it doesn’t matter which points you choose on the line,

the slope will work out to the same thing.

Example: Find the slope of the following lines.

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Example: Draw a line segment with a given slope.

2

3m =

3m = −

Example: Determine the slope of the line that passes through the following points.

(-5,-3) and (2,1)

(2,5) and (3, -2)

(1,-2) and (1,3)

(5,4) and (-2,4)

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Example: A line segment has endpoints at P(-3,2) and Q(x,5). Line segment PQ has a slope of 3.

Determine the value of x.

Example: Tom has a part-time job. He recorded the hours he worked and his pay for 3 different

days. On the first day he worked 2 hours and earned $24. On the second day he worked 4 hours and

made $48 and on the third day he worked 6 hours and made $72.

a) Plot these points on the grid below.

b) What is the slope of the line through these points?

c) What does the slope represent?

d) How can the answer to part c be used to determine:

a. How much Tom earned in 3.5 hours?

b. The time it took Tom to earn $30?

Assignment: Textbook page 339 #4-9, 13-17, 23, 24, 28

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Math 10C

Unit: Linear Functions

Topic: Slopes of parallel and perpendicular lines

Objectives:

• Use slope to determine if two lines are parallel, perpendicular or neither.

NOTES

When two lines have the same slope, congruent triangles can be drawn to show that the rise and run are

equal. So lines that have the same slope are parallel. (Parallel lines never intersect or they are

consistently the same distance away from each other).

Perpendicular lines are lines that intersect at right angles. When lines are perpendicular the slopes are

negative reciprocals of each other. A negative reciprocal is the value that would provide a product of

-1. For example, 1

3 13

− = − or 2 5

15 2

−= − . A slope of 0 is perpendicular to an undefined slope.

Example: Line EF passes through E(-3,-2) and F(-1,6). Line CD passes through C(-1,-3) and

D(1,7). Line AB passes through A(-3,7) and B(-5,-2). Sketch the lines. Are they parallel?

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Example: Line ST passes through S(-2,7) and T(2,-5). Line UV passes through U(-2,3) and

V(7,6). Are these two lines parallel, perpendicular, or neither? Sketch the lines to verify your answer.

Example: a) Determine the slope of a line that is perpendicular to the line through G(-2,3) and

H(1,2).

b) Determine the coordinates of J so that line GJ is perpendicular to line GH.

Example: EFGH is a parallelogram. Is it a rectangle?

E(-1,3) F(-3,-2) G(0,-3) H(2,2)

Example: Two lines have slopes of 3

4

− and

5

k . Find the value of k if the lines are parallel and if

they are perpendicular.

Assignment: Textbook page 349 #5, 6, 8-11, 13,16,20

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Math 10C

Unit: Linear Functions

Topic: Slope-Intercept form of a linear equation

Objectives:

• Relate the graph of a linear function to its equation in slope-intercept form.

NOTES

The equation of a linear function can be written in the form y mx b= + , where m is the slope and b is

the y-intercept (0,b).

Example: The graph of a linear function has slope -7/3 and a y-intercept at 5. Write the equation

for this function.

Example: Graph the linear function with equation 3

24

y x= + .

Example: Write and equation to describe this function. Verify the equation is correct.

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Example: To join the local gym, Jim pays a start-up fee of $99, plus a monthly fee of $29.

a) Write an equation for the total cost, C dollars, for n months at the gym.

b) Suppose Jim went to the gym for 23 months. What is the total cost?

c) Suppose the total cost was $505. How many months did Jim go to the gym?

d) Could the total cost ever be exactly $600?

Assignment: Textbook page 362 #4, 5, 7-9, 11-14, 18, 19, 21

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Math 10C

Unit: Linear Functions

Topic: Slope-Point form of a linear equation

Objectives:

• Relate the graph of a linear function to its equation in slope-point form.

NOTES

We saw that slope is calculated using the formula:

2 1

2 1

riseslope

run

ym

x

y ym

x x

=

=

−=

−

If we multiplied both sides of the equation by 2 1( )x x− , thus not changing the equation, only

rearranging it, we would have

2 12 1 2 1

2 1

2 1 2 1

( ) ( )

( )

y yx x m x x

x x

x x m y y

−− = −

−

− = −

This equation is called the slope-point form of a non-vertical line through point 1 1( , )x y and is more

commonly written as 1 1( )x x m y y− = −

Example: a) Write the equation of a line through (-2, 5) with a slope of -3

b) Express the equation in slope-intercept form.

Example: Find the equation of a line passing through (1, 4) with a slope of 3.

Example: Write the equation of the line that passes through (-5, -1) with a slope of 2/3

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Example: Determine the equation of a line passing through (5, 4) with a slope of zero

Example: Use slope-point form to write an equation of the line through (3, -4) and (5, -1). Verify

that the equation is correct.

Example: Determine the equation of the line graphed below.

Assignment: Page 372 #4abf, 5cd, 7bc, 8, 9i ii, 10, 11ab, 13, 15, 18, 19, 22, 25

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Math 10C

Unit: Linear Functions

Topic: General form of a linear equation

Objectives:

• Relate the graph of a linear function to its equation in general form.

NOTES

The general form of a linear equation is 0Ax By C+ + = , where A is a whole number and B and C

are integers. You can convert a linear equation from one form to another by applying the rules of

algebra.

Example: Convert 2

63

y x−

= + into general form.

Example: Convert 4 2 0x y− + = into slope-intercept form.

Example: Convert 3

4 ( 3)2

y x+ = − into general form.

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Finding x and y intercepts

We know that at the x-intercept the value of y is zero. Therefore if we are looking for the x-intercept

we can substitute 0 into our equation in the place of y. We are left with only the variable x in our

equation and we can algebraically solve for x giving us the x-intercept.

The same is true for the y-intercept. At the y-intercept the value of x is 0. By substituting zero into the

equation for x, we can solve for y to give us the y-intercept.

Example: Answer the following questions for the linear equation 2 3 6 0x y− − =

a) State the x-intercept of the graph.

b) State the y-intercept of the graph

c) Use the intercepts to graph the line.

Example: Given the linear equation 3 4 12 0x y+ + = , find the slope and the x and y intercepts.

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Example: When an equation is written in general form, what is the effect on the graph if A=0? What if

B=0?

Assignment: Page 384 #4, 5ab, 6cd, 7a, 8i ii, 10, 12ac, 13cd, 16, 18ac, 21, 23, 26, 27