How can we use slope?
There are a few ways to find slope. You can find it by using a graph or by using a formula.
Slope is calculated by finding the ratio that compares the vertical change to the horizontal change and writing it as a ratio of change in y over the change in x.
When finding slope from a graph we identify two coordinates, preferably two that are easy to work with and are integers. We can then count the rise and run from one point to the other! Finish by writing it as a ratio of change in y over the change in x.
If you are given two points that are on the line, and , you can find the slope by using the following formula:
Special Cases:
- When the line is vertical, then the slope is undefined since there is no change in the x-values.
- When the line is horizontal, then the slope is zero since there is no change in the y-values.
The sign of the slope can tell you the direction that the line will be. If the slope is positive, then the graph is rising or increasing from left to right. If the slope is negative, then the line is falling or decreasing from left to right.
Slope is also found in many real-world applications. As long as there is a constant rate of change, we can use linear equations to represent the situations.
Sample Math Problems
1. Find the slope of the line
2. Find the slope of the line
3. Find the slope of the line through the given pair of points.
Solution:
4. A local band will be performing at the stadium and must pay a fee of $1200 to rent the venue. The band will receive 40% of each $20 for each ticket that is sold. The maximum capacity of the stadium is 900 people.
a. Write an equation to represent the band’s revenue R when x tickets are sold.
Solution:
The slope is a rate of change while the y-intercept is a one-time occurrence. In this situation, the slope would be the 40% for each $20 ticket sold or 0.40 x 20 = 8. Be sure to write this slope as a decimal. Finally, the fee is a cost to the band, therefore it would be written as a negative value.
R = 8x - 1200
b. How much would the band’s revenue be if they sold out the concert?
Solution:
The maximum capacity was 900, therefore x = 900
R = 8(900) - 1200
R = 7200 - 1200
