Linear FunctionsCHAPTER 6

Chapter 66.1 – SLOPE OF A

LINE

SLOPE – STEEPNESS

Some roofs are steeper than others. Steeper roofs are more expensive to shingle.

So, what are the slopes for these three roofs?

EXAMPLE

Determine the slope of each line segment.

a) b)

TRY IT

Determine the slope of each line segment.

a) b)

HORIZONTAL/VERTICAL LINES

For a horizontal line segment: What’s the rise? What’s the run? For a vertical line

segment: What’s the

rise? What’s the

run?

Can you divide by zero?

EXAMPLE

Draw a line segment with each given slope.

a) b)

a) rise

run

b) rise

run

Independent practice

P. 339-340, #5, 6, 7, 9, 11

EXAMPLE

Determine the slope of the line that passes through C(–5, –3) and D(2, 1).

FORMULA

I would encourage everyone to solve for slope using the simple “rise over run” formula, and drawing a picture when necessary. However, there is a formula for those of you who like them.

A line passes through A(x1,y1) and B(x2,y2).

TRY IT

Determine the slope of the line that passes through E(4, –5) and F(8, 6).

EXAMPLE

Yvonne recorded the distances she had 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:

i) how far Yvonne travelled in 1.75 hours?ii) the time it took Yvonne to travel 55km?

Independent practice

PG. 340-343, #13, 16, 17, 18, 20, 25,

30.

Chapter 6

6.2 – SLOPES OF PARALLEL AND

PERPENDICULAR LINES

PARALLEL LINES

What does parallel mean?

Parallel lines are two lines on a plane that never meet, and are always the same distance apart.

What is the slope of these two lines?

EXAMPLE

A line GH passes through G(–4, 2) and H(2, –1). Line JK passes through J(–1, 7) and K(7,3). Line MN passes through M(–4, 5) and N(5, 1). Sketch the lines. Are they parallel? Justify your answer.

To show that lines are parallel, you need to show that they have the same slopes.

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?

CHALLENGE

Are these lines parallel?Show work.

PERPENDICULAR LINES

Non-parallel lines in the same plane have different slopes. Perpendicular lines are not parallel, so they have different slopes. What does perpendicular mean?

Perpendicular lines are lines that form a 90º angle where they meet.

The slopes of perpendicular lines are negative reciprocals of one another.

EXAMPLE

Line PQ passes through P(–7, 2) and Q(–2, 10). Line RS passes through R(–3, –4) and S(5, 1). a) Are these two lines parallel, perpendicular, or neither? Justify.b) Sketch the lines to verify the answer to part A.

Line ST passes through S(–2, 7) and T(2, –5). Line UV passes through U(–2, 3) and V(7, 6). Are these lines parallel, perpendicular, or neither?

EXAMPLE

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.

What is the slope of EF?

Draw it out:Use the formula:

What’s the negative reciprocal of it’s slope?

EXAMPLE

ABCD is a parallelogram. Is it a rectangle?Justify your answer.

Independent practice

PG. 349-351, #5, 6, 8, 9, 12, 13, 17, 19.

Chapter 66.3 – INVESTIGATING

GRAPHS OF LINEAR FUNCTIONS

CHALLENGE

ABCD is a parallelogram. Three vertices have coordinates A(–4, 3), B(2, 4), and C(4, 0). a) Is ABCD a rectangle? Justify your

answer. b) Determine the coordinates of D.

LINEAR FUNCTIONS

Alimina purchased an mp3 player and downloaded 3 songs. Each subsequent day, she downloads 2 song. Which graph represents this situation? Explain your choice.

Independent practicePG. 356, #3-6

Chapter 6

6.4 – SLOPE-INTERCEPT FORM OF THE EQUATION FOR

A LINEAR FUNCTION

LINEAR FUNCTIONS

This graph shows a cyclist’s journey where the distance is measured from her home.

What does the vertical intercepts represent?

What does the slope of the line represent?

SLOPE-INTERCEPT FORM

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

EXAMPLE

The graph of a linear function has slope and y-intercept –4. Write an

equation.

The graph of a linear function has

slope and y-intercept 5. Write an

equation.

Write the equation for this graph.

EXAMPLE

Graph the linear function with equation:

rise

run

slope = ½y-intercept = 3

Graph:

EXAMPLE

Write and equation to describe this function. Verify by checking a point on the graph.

EXAMPLE

The student council put on a dance. A ticket cost $5 and the cost for the DJ was $300.a) Write an equation for the profit, P dollars, on the sale of t 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.

Independent practice

PG. 362-364, #4, 5, 7, 8, 12, 13, 16, 18,

19.

Chapter 6

6.5 – SLOPE-POINT FORM OF THE

EQUATION FOR A LINEAR FUNCTION

POINT-SLOPE FORM

The line has slope –3 and passes through P(–2, 5). We use any other point Q(x, y) on the line to write an equation for the slope, m:

This form is called slope-point form. It’s used when you have the slope and a point, but not the y-intercept.

SLOPE-POINT FORM

The equation of a line that passes through P(x1,y1) and has slope m is:

EXAMPLE

a) Describe the graph of the linear function with this equation:

b) Graph the equation.

a)

What’s it slope?

What point does it go through?

b)

Remember that the x-value of the point is always the opposite sign.

EXAMPLE

a) Write an equation in slope-point form for this line.b) Write the equation in part a in slope-intercept form. What is the y-

intercept of this line?

We can’t see the y-intercept, so we need to use point-slope form.

What’s the slope?

What’s the point (x1,y1)?

EXAMPLE

a) Write an equation in slope-point form for this line.b) Write the equation in part a in slope-intercept form. What is the y-

intercept of this line?

b)

EXAMPLE

Write an equation for the line that passes through R(1, –1) and is:

a) Parallel to the line

b) Perpendicular to the line

Independent practice

PG. 372-374, #4, 7, 9, 11, 14, 17, 20, 23

CHALLENGE

Find the equation of the line with a slope of –1/3 going through point (–4,7)

Chapter 6

6.6 – GENERAL FORM OF THE EQUATION

FOR A LINEAR RELATION

GENERAL FORM

Ax + By + C = 0 is the general form of the equation of a line, where A is a whole number, and B and C are integers.

EXAMPLE

Write each equation in general form.

a) b)

a) Firstly, we want whole numbers, so we need to multiply by the denominator.

b) Multiply by the denominator.

Try these:

CHALLENGE

Write the following equation in both slope-intercept form and in general form:

EXAMPLE

a) Determine the x- and y-intercepts of the line whose equation is: 3x + 2y – 18 = 0

b) Graph the line.c) Verify that the graph is correct.

a) To determine the x-intercept, let y = 0

3x + 2y – 18 = 03x + 2(0) – 18 = 03x – 18 = 03x = 18x = 6

To find the y-intercept, let x = 0

3x + 2y – 18 = 03(0) + 2y – 18 = 02y – 18 = 02y = 18y = 9

EXAMPLE

a) Determine the x- and y-intercepts of the line whose equation is: 3x + 2y – 18 = 0

b) Graph the line.c) Verify that the graph is correct.

We found that:

x = 6y = 9

EXAMPLE

Determine the slope of the line with this equation: 3x – 2y – 16 = 0

Peanuts cost $2 per 100 g and raising cost $1 per 100 g. Devon has $10 to purchase both these items.a) Generate some data for this relation.b) Graph the data.c) Write an equation for the relation in general form.d) Will Devon spend exactly $10 if she buys 300 g of peanuts and 400 g of

raisins?e) Will Devon spend exactly $10 if she buys 400 g of peanuts and 300 g of

raisins?Use the graph and the equation to justify the answers.a) If she bought only peanuts, how many grams could

she buy?500 g

If she bought only raisins, how many grams could she buy?

1000 g

If she bought 200 g of peanuts, how many raisins could she buy?

200 g cost $4, so she could buy 600 g of raisins

Peanuts cost $2 per 100 g and raising cost $1 per 100 g. Devon has $10 to purchase both these items.a) Generate some data for this relation.b) Graph the data.c) Write an equation for the relation in general form.d) Will Devon spend exactly $10 if she buys 300 g of peanuts and 400 g of

raisins?e) Will Devon spend exactly $10 if she buys 400 g of peanuts and 300 g of

raisins?Use the graph and the equation to justify the answers.

Independent practice

PG. 384-385, #8, 9, 10, 12, 13, 16, 17,

18, 24