slope

(noun)

The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.

Related Terms

Examples of slope in the following topics:

  • Slope

    • Slope is often denoted by the letter $m$.
    • In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
    • The slope of the line is $\frac{4}{5}$.
    • We can see the slope is decreasing, so be sure to look for a negative slope.
    • Calculate the slope of a line using "rise over run" and identify the role of slope in a linear equation

  • Slope-Intercept Equations

    • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  
    • Let's write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept.
    • Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.
    • The value of the slope dictates where to place the next point.
    • The slope is $2$, and the $y$-intercept is $-1$.  

  • Point-Slope Equations

    • The point-slope equation is another way to represent a line; only the slope and a single point are needed.
    • Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
    • Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form
    • Since we have two points, but no slope, we must first find the slope:
    • Plug this point and the calculated slope into the point-slope equation to get:

  • Parallel and Perpendicular Lines

    • Recall that the slope-intercept form of an equation is: $y=mx+b$ and the point-slope form of an equation is: $y-y_{1}=m(x-x_{1})$, both contain information about the slope, namely the constant $m$.
    • This means that if the slope of one line is $m$, then the slope of its perpendicular line is $\frac{-1}{m}$.
    • The two slopes multiplied together must equal $-1$.
    • Also, the product of the slopes equals $-1$.
    • Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.  

  • What is a Linear Function?

    • Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
    • For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  
    • Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.  
    • Horizontal lines have a slope of zero and is represented by the form, $y=b$, where $b$ is the $y$-intercept.  
    • The blue line has a positive slope of $\frac{1}{2}$ and a $y$-intercept of $-3$; the red line has a negative slope of $-1$ and a $y$-intercept of $5$.

  • Difference Quotients

    • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
    • The difference quotient is the average slope of a function between two points.
    • However, it is important to note it is not necessarily to actual slope of the curve, as can be visually seen in .
    • To show how the above equation can be written as the average slope, put it into more familiar terms.
    • The difference quotient can be used to calculate the average slope (here, represented by a straight line) between two points P and Q.

  • Solving Systems Graphically

    • To do this, you need to convert the equations to slope-intercept form, or $y=mx+b$, where m = slope and b = y-intercept.
    • The best way to convert an equation to slope-intercept form is to first isolate the y variable and then divide the right side by B, as shown below.
    • Now $\displaystyle -\frac{A}{B}$ is the slope m, and $\displaystyle \frac{C}{B}$ is the y-intercept b.
    • Once you have converted the equations into slope-intercept form, you can graph the equations.

  • What is a Quadratic Function?

    • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope).
    • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

  • Linear and Quadratic Equations

    • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
    • These equations are often referred to as the "equations of the straight line. " In what follows, $x$, $y$, $t$, and $\theta$ are variables, $m$ is the slope, and $b$ is the y-intercept.
    • If $B$ is nonzero, then the y-intercept, or the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\displaystyle \frac{C}{B}$, and the slope of the line is $\displaystyle -\frac{A}{B}$.
    • Vertical lines, having undefined slopes, cannot be represented by this form.
    • The point-slope form expresses the fact that the difference in the y-coordinate between two points on a line (that is, $y-y_1$) is proportional to the difference in the x-coordinate (that is, $x-x_1$).

  • Zeroes of Linear Functions

    • Finally, if the line is vertical or has a slope, then there will be only one zero.
    • All lines, with a value for the slope, will have one zero.  
    • Since each line has a value for the slope, each line has exactly one zero.