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# Calculus Graphs Study Guide (page 3)

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## Straight Lines

The easiest and most beloved of all graphs are straight lines. Human beings tend to build, move, and even think in straight lines. There is something calming and reassuring about straight lines. With any two points, we can immediately tell how much a line is increasing or decreasing, as seen in Figure 2.7.

"How much a line is increasing or decreasing" is called the slope and is calculated by dividing "rise over run":

#### Example

What is the slope of the line through points (2,7) and (–1,5)?

## Point-Slope Formula

The most wonderful thing about straight lines is that their slopes are always the same. Thus, if a straight line has slope m and goes through the point (x1, y1) , then any other point (x,y) on the line will calculate the same slope:

By cross-multiplying, we get the point-slope formula for finding the equation of a straight line:

yy1 = m( xx1)

or equivalently

y = m(xx1) + y1

Here, y is a function of x, which could be written as

y(x) = m(xx1) + y1.

#### Example 1

Find the equation of the line with slope –2 through point (–1,8). Graph the line.

#### Solution 1

y= –2(x– (–1)) + 8

y = –2x + 6

This form of the equation is called the slope-intercept form because –2 is the slope and 6 is where the line intercepts the y-axis (see Figure 2.8):

The slope of –2 = means the y-value goes down 2 with every 1 increase in the x-value.

#### Example 2

Find the equation of the straight line through (2,6) and (5,7). Graph the line.

#### Solution 2

The slope is , so the equation is (see Figure 2.9).

The slope of means the y-value goes up when the x-value increases by 3.

Find practice problems and solutions for these concepts at Calculus Graphs Practice Questions.

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