Lines with Zero Slope and Undefined Slope

Example Use slope-intercept form to graph f (x) = 2

SolutionWriting in slope-intercept form: f (x) = 0 • x + 2. No matter what number we choose for x, we find that y must equal 2.

Choose any number for x

y must always be 2

x y (x, y)

0 2 (0, 2)

4 2 (4, 2)

4 2 (4 , 2)

f (x) = 2

Example continued y = f (x) = 2

When we plot the ordered pairs (0, 2), (4, 2) and (4, 2) and connect the points, we obtain a horizontal line.

Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis with y-intercept (0, 2).

x y (x, y)

2 4 (2, 4)

2 1 (2, 1)

2 4 (2, 4)

x must be 2

Example Graph x = 2

SolutionWe regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column.

Any number can be used for y

x = 2

Example continued x = 2

When we plot the ordered pairs (2, 4), (2, 1), and (2, 4) and connect them, we obtain a vertical line.

Any ordered pair of the form (2, y) is a solution. The line is parallel to the y-axis with x-intercept (2, 0).

Example

Find the slope of each given line. a. 4y + 3 = 15 b. 5x = 20

Solutiona. 4y + 3 = 15 (Solve for y)

4y = 12 y = 3

Horizontal line, the slope of the line is 0. b. 5x = 20 x = 4 Vertical line, the slope is undefined.

Graphing Using Intercepts

The point at which the graph crosses the x-axis is called the x-intercept. The y-coordinate of a x-intercept is always 0, (a, 0) where a is a constant.

The point at which the graph crosses the y-axis is called the y-intercept. The x-coordinate of a y-intercept is always 0, (0, b) where b is a constant.

x-intercepts

(3, 0)(-1, 0)

y-intercept

(0, -3)

Example Graph 5x + 2y = 10 using intercepts.

SolutionTo find the y-intercept we let x = 0 and solve for y. 5(0) + 2y = 10

2y = 10 y = 5 (0, 5)To find the x-intercept we let y = 0 and solve for x. 5x + 2(0) = 10

5x = 10 x = 2 (2, 0)

5x + 2y = 10

x-intercept (2, 0)

y-intercept (0, 5)

Go to your y-editor and enter the equation.

Select menu 1

Enter 0

So, the y-intercept is (0, -4).

Press

Press

Press

Press

Select 2 from the menu

Enter 3

Press

Enter 8

Press

Press

So, the x-intercept is (6, 0).

Press

Press

5Change window size to [ 10,5] [ 5,20]

Press

The three amigos use the notation

[ 10,5, 5,20] for window size.

Parallel and Perpendicular Lines

Two lines are parallel if they lie in the same plane and do not intersect no matter how far they are extended. If two lines are vertical, they are parallel.

Example Determine whether the graphs of and 3x 2y = 5 are parallel.

Solution When two lines have the same slope but different y-intercepts they are parallel.

The line has slope 3/2 and y-intercept (0, 3).

Rewrite 3x 2y = 5 in slope-intercept form:3x 2y = 5

2y = 3x 5

The slope is 3/2 and the y-intercept is (5/2, 0).

Both lines have slope 3/2 and different y-intercepts, the graphs are parallel.

33

2y x

33

2y x

3 52 2

y x

Example Determine whether the graphs of 3x 2y = 1 and are perpendicular.

Solution First, we find the slope of each line. The slope is – 2/3.

Rewrite the other line in slope-intercept form.

2 1

3 3y x

2

3

1

3y x

3 2 1x y

2 1 3y x

1 3

2 2

xy

1

2

3

2y x

The slope of the line is 3/2. The lines are perpendicular if the product of their slopes is 1.

2 31

3 2

The lines are perpendicular.

Example

Write a slope-intercept equation for the line whose graph is described.a) Parallel to the graph of 4x 5y = 3, with y-intercept (0, 5).b) Perpendicular to the graph of 4x 5y = 3, with y-intercept (0, 5).Solution (a) We begin by determining the slope of the line.

4 5 3x y

5 4 3y x 4 3

5 5y x The slope is

4.

5

A line parallel to the graph of 4x 5y = 3 has a slope of Since the y-intercept is (0, 5), the slope-intercept equation is

45 .

45

5y x

continued

b) A line perpendicular to the graph of 4x 5y = 3 has a slope that is the negative reciprocal of Since the y-intercept is (0, 5), the slope-intercept equation is

545 4or .

55

4y x

Press

They don’t appear to bePerpendicular.

Press

Select 5 from the menu

That’s better!

Recognizing Linear Equations

A linear equation may appear in different forms, but all linear equations can be written in standard form Ax + By = C.

Example

Determine whether each of the following equations is linear: a) y = 4x + 2 b) y = x2 + 3

Solution Attempt to write each equation in standard form. a) y = 4x + 2 4x + y = 2 Adding 4x to both sides

b) y = x2 + 3 x2 + y = 3 Adding x2 to both sides The equation is not linear since it has an x2 term.