How to Understand Slope (in Algebra)
Define slope., Visualize a line’s “rise over run.” The term “rise” refers to the vertical distance between two points, or the change in y{\displaystyle y}., Locate the slope of a line in an equation., Assess the steepness of the line., Identify a...
Step-by-Step Guide
-
Step 1: Define slope.
The slope is a measure of how steep a straight line is.A variety of branches of mathematics use slope.
In geometry, you can use the slope to plot points on a line, including lines that define the shape of a polygon.
Statisticians use slope to describe the correlation between two variables.Economists use slope to show and predict rates of change.People also use slope in real, concrete ways.
For example, slope is used when constructing roads, stairs, ramps, and roofs. -
Step 2: Visualize a line’s “rise over run.” The term “rise” refers to the vertical distance between two points
The term “run” refers to the horizontal distance between two points, or the change in x{\displaystyle x}.
When learning about the slope of a line, you will often see the formula slope=riserun{\displaystyle {\text{slope}}\;={\frac {\text{rise}}{\text{run}}}}For example a slope of a line might be 21{\displaystyle {\frac {2}{1}}}.
This means that to go from one point to the next, you need to go up 2 along the y-axis, and over 1 along the x-axis. , You can do this using the slope-intercept form of a line’s equation.
The slope-intercept form says that y=mx+b{\displaystyle y=mx+b}.
In this formula, m{\displaystyle m} equals the slope of the line.
You can rearrange the equation of a line into this formula to find the slope.For example, in the equation y=3x+1{\displaystyle y=3x+1}, the slope would be 3{\displaystyle 3}.
You can still think of this slope in terms of rise over run if you turn it into a fraction.
Any whole number can be turned into a fraction by placing it over
1.
So, 3=31{\displaystyle 3={\frac {3}{1}}}.
This means that the line represented by this equation rises 3 units vertically for every 1 unit it runs horizontally. , The larger the slope, the steeper the line.
A line is steeper the more vertical it rests on a coordinate plane.For example, a slope of 2 (that is, 21{\displaystyle {\frac {2}{1}}}) is steeper than a slope of
0.5 (12{\displaystyle {\frac {1}{2}}}). , A positive slope is one that moves up and to the right.
In other words, in a positive slope, as x{\displaystyle x} increases, y{\displaystyle y} also increases.
A positive slope is denoted by a positive number. , A negative slope is one that moves down and to the right.
In other words, in a negative slope, as x{\displaystyle x} increases, y{\displaystyle y} decreases.
A negative slope is denoted by a negative number, or a fraction with a negative numerator.
To help remember the difference between a positive and negative slope, you can think of yourself as standing on the left endpoint of the line.
If you need to walk up the line, it’s positive.
If you need to walk down the line, it’s negative.Knowing the difference between negative and positive slopes can help you check that your calculations are reasonable. , A horizontal line is a line that runs straight across a coordinate plane.
The slope of a horizontal line is
0.
This makes sense if you think of lines in terms of slope=riserun{\displaystyle {\text{slope}}\;={\frac {\text{rise}}{\text{run}}}}.
For a horizontal line, the rise is 0, since the y{\displaystyle y} value never increases or decreases.
So, the slope of a horizontal line would be 0x{\displaystyle {\frac {0}{x}}}. , The slope of a vertical line is undefined.
In terms of riserun{\displaystyle {\frac {\text{rise}}{\text{run}}}}, the slope of a negative line would be y0{\displaystyle {\frac {y}{0}}}.
The run is 0, since the x{\displaystyle x} value never increases or decreases.
So, the slope of a vertical line will bey0{\displaystyle {\frac {y}{0}}}, and since you can't divide by 0, any number over 0 will always be undefined. -
Step 3: or the change in y{\displaystyle y}.
-
Step 4: Locate the slope of a line in an equation.
-
Step 5: Assess the steepness of the line.
-
Step 6: Identify a positive slope.
-
Step 7: Identify a negative slope.
-
Step 8: Understand the slope of a horizontal line.
-
Step 9: Understand the slope of a vertical line.
Detailed Guide
The slope is a measure of how steep a straight line is.A variety of branches of mathematics use slope.
In geometry, you can use the slope to plot points on a line, including lines that define the shape of a polygon.
Statisticians use slope to describe the correlation between two variables.Economists use slope to show and predict rates of change.People also use slope in real, concrete ways.
For example, slope is used when constructing roads, stairs, ramps, and roofs.
The term “run” refers to the horizontal distance between two points, or the change in x{\displaystyle x}.
When learning about the slope of a line, you will often see the formula slope=riserun{\displaystyle {\text{slope}}\;={\frac {\text{rise}}{\text{run}}}}For example a slope of a line might be 21{\displaystyle {\frac {2}{1}}}.
This means that to go from one point to the next, you need to go up 2 along the y-axis, and over 1 along the x-axis. , You can do this using the slope-intercept form of a line’s equation.
The slope-intercept form says that y=mx+b{\displaystyle y=mx+b}.
In this formula, m{\displaystyle m} equals the slope of the line.
You can rearrange the equation of a line into this formula to find the slope.For example, in the equation y=3x+1{\displaystyle y=3x+1}, the slope would be 3{\displaystyle 3}.
You can still think of this slope in terms of rise over run if you turn it into a fraction.
Any whole number can be turned into a fraction by placing it over
1.
So, 3=31{\displaystyle 3={\frac {3}{1}}}.
This means that the line represented by this equation rises 3 units vertically for every 1 unit it runs horizontally. , The larger the slope, the steeper the line.
A line is steeper the more vertical it rests on a coordinate plane.For example, a slope of 2 (that is, 21{\displaystyle {\frac {2}{1}}}) is steeper than a slope of
0.5 (12{\displaystyle {\frac {1}{2}}}). , A positive slope is one that moves up and to the right.
In other words, in a positive slope, as x{\displaystyle x} increases, y{\displaystyle y} also increases.
A positive slope is denoted by a positive number. , A negative slope is one that moves down and to the right.
In other words, in a negative slope, as x{\displaystyle x} increases, y{\displaystyle y} decreases.
A negative slope is denoted by a negative number, or a fraction with a negative numerator.
To help remember the difference between a positive and negative slope, you can think of yourself as standing on the left endpoint of the line.
If you need to walk up the line, it’s positive.
If you need to walk down the line, it’s negative.Knowing the difference between negative and positive slopes can help you check that your calculations are reasonable. , A horizontal line is a line that runs straight across a coordinate plane.
The slope of a horizontal line is
0.
This makes sense if you think of lines in terms of slope=riserun{\displaystyle {\text{slope}}\;={\frac {\text{rise}}{\text{run}}}}.
For a horizontal line, the rise is 0, since the y{\displaystyle y} value never increases or decreases.
So, the slope of a horizontal line would be 0x{\displaystyle {\frac {0}{x}}}. , The slope of a vertical line is undefined.
In terms of riserun{\displaystyle {\frac {\text{rise}}{\text{run}}}}, the slope of a negative line would be y0{\displaystyle {\frac {y}{0}}}.
The run is 0, since the x{\displaystyle x} value never increases or decreases.
So, the slope of a vertical line will bey0{\displaystyle {\frac {y}{0}}}, and since you can't divide by 0, any number over 0 will always be undefined.
About the Author
Cheryl Russell
Enthusiastic about teaching pet care techniques through clear, step-by-step guides.
Rate This Guide
How helpful was this guide? Click to rate: