Section 1.4 LinesObjectives:1. Calculate and interpret the slope of a line.2. Graph lines given a point and the slope.3. Find the equation of a vertical line.4. Use the point-slope Form of a line; Identify

Horizontal Lines5. Find the equation of the line given two points.6. Write the equation in slope-intercept form.7. Identify the slope and y-intercept of a line from

its equation.8. Graph lines written in general form using

intercepts.9. Find equations of parallel lines.10.Find equations of perpendicular lines.

11 y ,x 22 y ,x

1 2x = x

21 xx

Calculating Slope

The slope of the line above appears to about 1. How hard would SkiBird have to work if the slope was larger than 1?What if it was less than 1?

Calculating Slope

Negative SlopeLines that have a negative slope, slant “downhill” (as viewed from

left to right).

SkiBird enjoys the ride down the hill. He needs to occasionally use

negative energy to try to slow down.

The slope of the line above appears to be about -1.How fast would SkiBird be going if the slope was -3?What about -1/2?

Calculating Slope

Calculating Slope

Zero SlopeLines that are horizontal have

zero slope.

SkiBird is cross-country skiing on level ground. He is not

working hard to get up a hill, nor trying to slow down. His energy level (and enjoyment

level) is at 0.

Calculating Slope

As you can see from the diagram, another way we talk about slope is rise over run. The rise represents the change in y, sometimes symbolized by ∆y. The run represents the change in x, sometimes symbolized by ∆x.

The slope m of a nonvertical line L measures the amount that y changes as x changes from . We call this the average rate of change of y with respect to x.

21 xto x

Calculating Slope

Calculate the slope of the line graphed below.

Calculating Slope

Calculate the slope of the line using the table.X Y

0 5

5 30

10 55

15 80

Interpreting Slope

The slope of a line was found to be 4/5. How would you interpret the slope?

Answer:

Remember the slope represents the change in y over the change in x. So, a line with a slope of 4/5 means….for every 5-unit change in x, y will change by 4 units. Or, if x increases 5 units, y will increase 4 units. The average rate of change of y with respect to x is 4/5.

Interpreting Slope

Look at the graph to the left. Estimate the slope of the line. Then interpret its meaning in the context of this graph.

Graph the lineGraph the line that contains the point (3, 2) and has a slope of 2.Graph the line that contains the point (3, 2) and has a slope of -1/2 .

Find the equation of a vertical line.

Graph the line x = 3.To graph x = 3 by hand, we are looking for all points in the plane for which x = 3. No matter what y-coordinate is used the corresponding x-coordinate will be 3. Consequently, the graph of the equation x = 3 is a vertical line.

A vertical line is given by an equation of the form

x = awhere a is the x-intercept.

Use the point-slope form of a line

An equation of a nonvertical line of slope m that contains the point (x1, y1 ) is

Y – y1 = m (x – x1)

Find the equation of the line with slope 4 and containing the point (1, 2).

Find the equation of the line with slope -3 and containing the point (-2, 5).

Use the point-slope form of a line

Find the equation of the line whose slope is 0 and containing the point (4, -3).

Equations of horizontal lines

A horizontal line is given by an equation of the form

y = b

where b is the y-intercept.

Use the point-slope form of a lineFind the equation of the line that contains the points (2, 3) and (-4, 5). Graph the line.

Use the point-slope form of a lineFind the equation of the line that contains the points (1, 3) and (-1, 2). Graph the line.

Slope Intercept Form of a Line

An equation of a line with slope m and y-intercept b is y = mx + b

Change the following equations to slope-intercept form. Then identify the slope and y-intercept.

y – 5 = 3 (x + 4)

2x + 4y = 8

Slope Intercept Form of a Line

An equation of a line with slope m and y-intercept b is y = mx + b

Change the following equations to slope-intercept form. Then identify the slope and y-intercept.

y + 5 = -2 (x – 5)

-9x – 6y = 18

Graph lines written in general formThe equation of the line is in general form when it is written as Ax + By = C, where A, B, and C are real numbers and A and B are not both zero.When equations are given to you in general form it is easier to graph them by using their intercepts.Graph the equation 2x + 4y = 8.