The grade, or slope, of a road is measured in a percentage. For example, if a road has a downgrade of 7%, this means, that over every 100 horizontal feet, the road will slope down 7 feet vertically.
If a highway has a downgrade of 12% for 3 miles (5280 feet in a mile), how much will the road drop? What is the slope of this stretch of highway?
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Guidance
The slope of a line determines how steep or flat it is. When we place a line in the coordinate plane, we can measure the slope, or steepness, of a line. Recall the parts of the coordinate plane, also called a \begin{align*}x-y\end{align*} plane and the Cartesian plane, after the mathematician Descartes.
To plot a point, order matters. First, every point is written \begin{align*}(x, y),\end{align*} where \begin{align*}x\end{align*} is the movement in the \begin{align*}x-\end{align*}direction and \begin{align*}y\end{align*} is the movement in the \begin{align*}y-\end{align*}direction. If \begin{align*}x\end{align*} is negative, the point will be in the \begin{align*}2^{nd}\end{align*} or \begin{align*}3^{rd}\end{align*} quadrants. If \begin{align*}y\end{align*} is negative, the point will be in the \begin{align*}3^{rd}\end{align*} or \begin{align*}4^{th}\end{align*} quadrants. The quadrants are always labeled in a counter-clockwise direction and using Roman numerals.
The point in the \begin{align*}4^{th}\end{align*} quadrant would be (9, -5).
To find the slope of a line or between two points, first, we start with right triangles. Let’s take the two points (9, 6) and (3, 4). Plotting them on a \begin{align*}x-y\end{align*} plane, we have:
To turn this segment into a right triangle, draw a vertical line down from the higher point, and a horizontal line from the lower point, towards the vertical line. Where the two lines intersect is the third vertex of the slope triangle.
Now, count the vertical and horizontal units along the horizontal and vertical sides (\begin{align*}{\color{red}{\mathbf{red}}}\end{align*} dotted lines).
The slope is a fraction with the vertical distance over the horizontal distance, also called the “rise over run.” Because the vertical distance goes down, we say that it is -2. The horizontal distance moves towards the negative direction (the left), so we would say that it is -6. So, for slope between these two points, the slope would be \begin{align*}\frac{-2}{-6}\end{align*} or \begin{align*}\frac{1}{3}\end{align*}.
Note: You can also draw the right triangle above the line segment.
Example A
Use a slope triangle to find the slope of the line below.
Solution: Notice the two points that are drawn on the line. These are given to help you find the slope. Draw a triangle between these points and find the slope.
From the slope triangle above, we see that the slope is \begin{align*}\frac{-4}{4} = -1\end{align*}.
Whenever a slope reduces to a whole number, the “run” will always be positive 1. Also, notice that this line points in the opposite direction as the line segment above. We say this line has a negative slope because the slope is a negative number and points from the \begin{align*}2^{nd}\end{align*} to \begin{align*}4^{th}\end{align*} quadrants. A line with positive slope will point in the opposite direction and point between the \begin{align*}1^{st}\end{align*} and \begin{align*}3^{rd}\end{align*} quadrants.
If we go back to our previous example with points (9, 6) and (3, 4), we can find the vertical distance and horizontal distance another way.
From the picture, we see that the vertical distance is the same as the difference between the \begin{align*}y-\end{align*}values and the horizontal distance is the difference between the \begin{align*}x-\end{align*}values. Therefore, the slope is \begin{align*}\frac{6-4}{9-3}\end{align*}. We can extend this idea to any two points, \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2)\end{align*}.
Slope Formula: For two points \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2),\end{align*} the slope between them is \begin{align*}\frac{y_2 - y_1}{x_2 - x_1}\end{align*}. The symbol for slope is \begin{align*}m\end{align*}.
It does not matter which point you choose as \begin{align*}(x_1, y_1)\end{align*} or \begin{align*}(x_2, y_2)\end{align*}.
Example B
Find the slope between (-4, 1) and (6, -5).
Solution: Use the Slope Formula above. Set \begin{align*}(x_1, y_1) = (-4, 1)\end{align*} and \begin{align*}(x_2, y_2) = (6, -5)\end{align*}.
\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 -(-4)}{-5 - 1} = \frac{10}{-6} = -\frac{5}{3}\end{align*}
Example C
Find the slope between (9, -1) and (2, -1).
Solution: Use the Slope Formula. Set \begin{align*}(x_1, y_1) = (9, -1)\end{align*} and \begin{align*}(x_2, y_2) = (2, -1)\end{align*}.
\begin{align*}m = \frac{-1 - (-1)}{2 - 9} = \frac{0}{-7} = 0\end{align*}
Here, we have zero slope. Plotting these two points we have a horizontal line. This is because the \begin{align*}y-\end{align*}values are the same. Anytime the \begin{align*}y-\end{align*}values are the same we will have a horizontal line and the slope will be zero.
Intro Problem Revisit The road slopes down 12 feet over every 100 feet.
Let's set up a ratio to find out how much the road slopes in 3 miles, or \begin{align*}3 \cdot 5280 = 15,840\end{align*} feet.
\begin{align*}\frac{12}{100}&= \frac{x}{15,840} \\ 15840 \cdot \frac{12}{100} &= x \\ x &= 1900.8 \end{align*}
The road drops 1900.8 feet over the 3 miles. The slope of the road is \begin{align*}\frac{12}{100}\end{align*} or \begin{align*}\frac{3}{25}\end{align*} when the fraction is reduced.
Guided Practice
1. Use a slope triangle to find the slope of the line below.
2. Find the slope between (2, 7) and (-3, -3).
3. Find the slope between (-4, 5) and (-4, -1).
Answers
1. Counting the squares, the vertical distance is down 6, or -6, and the horizontal distance is to the right 8, or +8. The slope is then \begin{align*}\frac{-6}{8}\end{align*} or \begin{align*}-\frac{2}{3}\end{align*}.
2. Use the Slope Formula. Set \begin{align*}(x_1, y_1) = (2, 7)\end{align*} and \begin{align*}(x_2, y_2) = (-3, -3)\end{align*}.
\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 -7}{-3 -2} = \frac{-10}{-5} = 2\end{align*}
3. Again, use the Slope Formula. Set \begin{align*}(x_1, y_1) = (-4, 5)\end{align*} and \begin{align*}(x_2, y_2) = (-4, -1)\end{align*}.
\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 -5}{-4 -(-4)} = \frac{-6}{0}\end{align*}
You cannot divide by zero. Therefore, this slope is undefined. If you were to plot these points, you would find they form a vertical line. All vertical lines have an undefined slope.
Important Note: Always reduce your slope fractions. Also, if the numerator or denominator of a slope is negative, then the slope is negative. If they are both negative, then we have a negative number divided by a negative number, which is positive, thus a positive slope.
Explore More
Find the slope of each line by using slope triangles.
Find the slope between each pair of points using the Slope Formula.
- (-5, 6) and (-3, 0)
- (1, -1) and (6, -1)
- (3, 2) and (-9, -2)
- (8, -4) and (8, 1)
- (10, 2) and (4, 3)
- (-3, -7) and (-6, -3)
- (4, -5) and (0, -13)
- (4, -15) and (-6, -11)
- (12, 7) and (10, -1)
- Challenge The slope between two points \begin{align*}(a, b)\end{align*} and (1, -2) is \begin{align*}\frac{1}{2}\end{align*}. Find \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.