F - Slope of a Functions Lesson

Slope of a Functions

Let's talk slope... The "steepness" of a line. The slope of a line can be found directly when a linear equation is in slope-intercept form (y = mx + b). In this form, the slope is the coefficient of x and is represented by the letter m. The slope of a line can also be found by determining the ratio of the "rise" to the "run" between two points on the graph. In other words, slope measures how much the line rises vertically given a particular run or horizontal distance

If you have two points, you can substitute those into the formula for slope which is defined in the above paragraph. Slope is labeled with an "m." Here is the formula for slope.LaTeX: m=\frac{y_2-y_1}{x_2-x_1}m=y2−y1x2−x1

If you are given a graph, you have a couple of options for finding the slope, or rate of change. You can pick two points and use the above formula. Often times, slope is defined as rise/run, "rise over run," which is the same thing as the differences in y divided by the differences in x. If your graph is linear, meaning a line, counting the rise over run is a great method to check, or even find, the slope of the line. You could also, if given the function, put it in slope intercept form and pull the slope straight out of the equation.

What is slope intercept form? Let's take a look.

Slope intercept form:

y = mx + b         m is the slope and b is the y-intercept. The y-intercept is simply where the graph crosses the y axis. (when x = 0)

Example:

Let's compare the rates of change and the y-intercepts of two functions. 

  1.   y = -4x + 4
  2. A line that passes through the points (4,0) and (-4, 4).

The first thing we need to do is find the slope, or rate of change, for each.

Let's consider the first function, y = -4x + 4. This function is written in slope intercept form, so it is quite simple to pull the slope straight out of the equation. Remember, slope is represented by m in the formula y = mx + b. In this function, m = -4, so the slope is -4.

In the second function, we'll need to plot the points on the coordinate system and count the rise over run, or we can use the slope formula and the given points to calculate the slope. Let's use the latter.

LaTeX: m=\frac{y_2-y_1}{x_2-x_1}m=y2−y1x2−x1

m = 4 - 0
      -4 - 4

m = 4
      -8
m = - ½

We can conclude that the rates of change for the two functions are different.

The second part of the question is to compare the y-intercepts.

On the first function, we can again use the fact that it is written in slope-intercept form. The y-intercept is 4. This simply means the graph crosses the y axis at the point (0,4).

On the second function, we find ourselves with a couple of options. We can graph it and see where the graph crosses the y-axis or we can use the slope we just found and one of the points to put it in slope-intercept form so we can see the y-intercept. We will do the latter in this example.

y=mx+b y=(-1/2)x+b Substitute the slope, m, found earlier in the problem 0=(-1/2)(4)+b Use either of the original ordered pairs and substitute them in for x and y. 0=-2+b Add 2 to each side. 2=b Since b represents the y-intercept, we have found that the y-intercept is 2, which represents the point (0, 2). 

We can conclude that the y-intercepts for the two functions are different.

If the slope is positive, it means your function is increasing.

If the slope is negative, it means your function is decreasing.

If the slope is zero, it means you have a horizontal line.

If the slope is undefined (division by zero), it means you have a vertical line.

When determining is a function is increasing or decreasing, and on what interval, you'll want to read the graph from left to right. It may be helpful to put your finger on the left most part of the graph first. Run your finger along the graph. If your finger is moving upward, the graph is increasing. If you finger is moving download, then your graph is decreasing.

Example

image of parabola facing up Start with your finger at the purple arrow that says "start here". Move your finger along the graph. What happens? You are moving your finger downward until you get to the origin, the point (0, 0). This means that you left half of the graph is decreasing. When you get passed the point (0, 0), you are moving your finger up as you follow the graph. This means on the interval in the first quadrant, your function is increasing. You may have intervals of increase and intervals and decrease in the same graph.

Assignment

Now that you have spent some time learning about functions, you are ready to complete your Functions Homework. Click here to download the Functions Homework. Links to an external site.

Once you have completed the Functions Homework, check your even answers and make sure you ask your teacher if you have any questions. Links to an external site. When you feel confident in your work, take the Functions Homework Quiz. Make sure you have your Homework completed and with you when you log in to take it. This is a timed quiz and you'll only have time to enter the answers from your HW. You will not have time to work them out during the quiz.

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