Slope

Intro Have you ever pulled up to a garage and noticed the arm that restricts you from going in and/or out? Usually, you’ll see more than one car try to get through before the arm goes down and someone always gets clipped! The movement of arm changes from a horizontal, or level, position to a vertical position and then goes down again to a horizontal one. As the arm rises and lowers the steepness of the arm changes. The numeric steepness of the arm is referred to as the slope. Slope measures the steepness of the line as you look at it from left to right.

Finding the Slope One way of indicating the steepness of a line is by using an angle. The angle which is formed by the line and the positive x-axis is called the angle of inclination. For example, A (∠A) is the angle of inclination:

y

y

A

A x

x

The graph on the left has an angle of inclination which is acute causing the line to have a positive inclination. The graph of the right has an angle of inclination which is obtuse causing the line to have a negative inclination. A second way of indicating steepness is by using a number called a slope. This method is most commonly used when working with a coordinate system.

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Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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A line which rises as you move from left to right has a slope which is a positive number. A line which falls as you move from left to right has a slope which is a negative number. To determine the slope of a line, look at the ratio of corresponding y-values to x-values of the line. Let’s look at the following graphs:

2 units

4 units 5 units

2 units

Graph A

Graph B

To find the slope, first I locate two points on the line.

To find the slope, first I locate two points on the line.

Second, I count the number of units in the y-direction and then in the xdirection to get from one point to the other.

Second, I count the number of units in the y-direction and then in the xdirection to get from one point to the other.

Finally, I write the ratio of the change in the y-value over the change in the x-value.

Finally, I write the ratio of the change in the y-value over the change in the x-value.

In this example,

2units . 4 units

In this example,

5units 2units

.

As we mentioned before, the steepness of the line is determined by the value of the slope. Notice graph A has a slope of

2 5 and graph B has a slope of . The steeper the line, the larger 4 2

the slope and vice-versa.

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Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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Formula for Finding Slope Sometimes you will be given two points of a line and asked to determine the slope without being given a graph to look at. To solve this type of problem, you can use the formula for slope:

The slope of a line containing points (x1 , y1) and (x2 , y2) is m =

y2 − y1 . x2 − x1

Ok, so what does this all mean? Suppose you were given the following question on your exam: A line contains the points (3,4) and (5,8). What is the slope of the line? To solve this problem, we will use our formula for finding slope. Now, you could graph these two points on a coordinate system, draw the line that connects them and visually find the change-in-y and the change-in-x. However, this may take longer to answer. From the formula, if we find the ratio of the difference in y- and x-values, we can determine our slope. So,

m=

8−4 4 = =2 5−3 2

Tip: Notice I have reduced the ratio. It is not necessary to do so; however your answer choices may have the reduced form. Just keep that in mind.

When computing the slope of a line, remember to always use the change-in-y over the changein-x. Students often reverse the order and get the wrong solution. Your exam may include this mistake as one of your choices so be careful! For example, using the values above:

m=

5−3 2 1 = = 8 − 4 4 2 , as you can see if I flip-flop the change-in-y and change-in-x, I get a

different value. This value might actually appear as one of your choices (they try real hard to trick you!).

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Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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Types of Slope The slope of a line can be positive, negative, zero or undefined. The best way to illustrate different slopes is to see their graphs:

Here is an example of a positive slope.

Here is an example of a negative slope.

Notice the line is rising; if I were walking on the line from left-to-right, I would be traveling uphill.

Notice the line is falling; if I were walking on the line from left-to-right, I would be traveling downhill.

The slope of this line is

2 =1. 2

Note: I am determining my values from the red point to the blue point.

The slope of this line is

3 = −1 . −3

The bottom value is -3 because I moved in the negative direction on the x-axis. Note: I am determining my values from the red point to the blue point.

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Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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Here is an example of a zero slope.

Here is an example of an undefined slope.

Notice the line is level; if I were walking on the line from left-to-right, I would be traveling on level ground.

Notice the line is vertical; if I were walking on the line from left-to-right, I would be Spiderman.

The slope of this line is

0 =0. 4

The slope of this line is

4 = undefined . 0

The value is zero because a 0 divided by any number is always zero.

The value is undefined because you can not divide a value by zero.

Note: I am determining my values from the blue point to the red point.

Note: I am determining my values from the blue point to the red point.

Slope-Intercept Form So far, we have determined the slope of a line by simply counting the units on a coordinate system and by being given two points. Another way you may encounter slope-type questions is by being given an equation. The easiest way to determine the slope of an equation is by writing it in slope-intercept form:

y = mx + b Once an equation is written in this form, the slope will always be the value of m. The value of b determines the y-intercept (where the line intersects the y-axis).

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Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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3 4

1 2

Here’s an example: The equation of the line below is y = x − . Two points on line are (-2,-2) and (2,1). Using the slope formula:

m=

−2 − 1 −3 3 = = −2−2 −4 4

From the equation, I see the y-intercept 1 2

is − . In other words, the line intersects the y-axis at - 0.5.

In the example above, I calculated the slope to prove the m-value in the equation is in fact the slope of the equation. If you ever come across an equation of a line and you are trying to find its slope, simply write the equation in slope-intercept form and look at m.

Rewriting an Equation Here are some examples showing you how to convert an equation into slope-intercept form. Example 1 Rewrite the equation 2y = x + 8 to slope-intercept form. Solution 2y = x + 8 2y x 8 = + 2 2 2

1 y = x +4 2

Divide both sides by 2 to isolate y.

Tip: Remember, if you do not see a number (coefficient) in front of the variable x, we always assume it is a 1.

The slope of this equation is

4

1 . 2

Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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Example 2 Find the slope and y-intercept of the equation 4x + y = 3. Solution First, write the equation in slope-intercept form: 4x + y = 3 -4x

-4x

Subtract 4x from both sides to isolate y.

y = -4x + 3, the slope of this line is -4 and the y-intercept is 3.

Example 3 Find the slope and y-intercept of the equation 9x + 3y = 7. Solution First, write the equation in slope-intercept form: 9x + 3y = 7 -9x

-9x

Subtract 9x from both sides.

3y = -9x + 7 3y −9 7 = x+ 3 3 3

y = −3x +

Divide both sides by 3 to isolate y.

7 , the slope of this line is -3 3

and the y-intercept is

4

7 3

.

Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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Example 4 Rewrite the equation 4 x − 8y + 3 = 0 in slope-intercept form. 4x - 8y + 3 = 0 -4x -4x 8y + 3 = -4x -3 -3

Subtract 4x from both sides to isolate y.

Subtract 3 from both sides to isolate y.

8y = -4x – 3 8y −4 3 = x− 8 8 8

y=

4

Divide both sides by 8 to isolate y.

−1 3 x − , the slope of this line is −1 and the y-intercept is − 3 . 2 8 2 8

Knowledge of Geometry Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles solving problems.

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