The Slope and Equation of a Line Help
Introduction to The Slope and Equation of a Line
The slope of a line and the meaning of the slope are important in calculus. In fact, the slope formula is the basis for differential calculus. The slope of a line measures its tilt. The sign of the slope tells us if the line tilts up (if the slope is positive) or tilts down (if the slope is negative). The larger the number, the steeper the slope.
We can put any two points on the line, ( x 1 , y 1 ) and ( x 2 , y 2 ), in the slope formula to find the slope of the line.
A slope of means that if we increase the x -value by 2, then we need to increase the y -value by 1 to get another point on the line. For example, knowing that (0, 3) is on the line means that we know (0 + 2, 3 + 1) = (2, 4) is also on the line.
As we can see from Figure 1.4, (−4, −2) and (1, −2) are two points on a horizontal line. We will put these points in the slope formula.
Slopes of Horizontal and Vertical Lines
The slope of every horizontal line is 0. The y -values on a horizontal line do not change but the x -values do.
What happens to the slope formula for a vertical line?
The points (3, 2) and (3, − 1) are on the vertical line in Figure 1.5. Let’s see what happens when we put them in the slope formula.
This is not a number so the slope of a vertical line does not exist (we also say that it is undefined). The x -values on a vertical line do not change but the y -values do.
Any line is the graph of a linear equation. The equation of a horizontal line is y = a (where a is the y -value of every point on the line). Some examples of horizontal lines are y = 4, y = 1, and y = −5 .
The equation of a vertical line is x = a (where a is the x -value of every point on the line). Some examples are x = −3, x = 2, and x = 4.
Slope-Intercept and Point-Slope Form
Other equations usually come in one of two forms: Ax+By = C and y = mx+b . We will usually use the form y = mx + b in this book. An equation in this form gives us two important pieces of information. The first is m , the slope. The second is b , the y -intercept (where the line crosses the y -axis). For this reason, this form is called the slope-intercept form. In the line , the slope of the line is and the y -intercept is (0, 4), or simply, 4.
We can find an equation of a line by knowing its slope and any point on the line. There are two common methods for finding this equation. One is to put m, x , and y (x and y are the coordinates of the point we know) in y = mx + b and use algebra to find b. The other is to put these same numbers in the point-slope form of the line, y − y 1 = m ( x − x 1 ). We will use both methods in the next example.
Find an equation of the line with slope containing the point (8, −2). We will let m , and y = −2 in y = mx + b to find b.
The line is
Now we will let and y 1 = −2 in y − y 1 = m ( x − x 1 ).
- Find an equation of the line with slope 4, containing the point (0, 3). We know the slope is 4 and we know the y -intercept is 3 (because (0, 3) is on the line), so we can write the equation without having to do any work: y = 4 x + 3.
- Find an equation of the horizontal line that contains the point (5, −6). Because the y -values are the same on a horizontal line, we know that this equation is y = −6. We can still find the equation algebraically using the fact that m = 0, x = 5 and y = −6. Then y = mx + b becomes −6 = 0(5) + b. From here we can see that b = −6, so y = 0 x − 6, or simply, y = −6.
- Find an equation of the vertical line containing the point (10, −1). Because the x -values are the same on a vertical line, we know that the equation is x = 10. We cannot find this equation algebraically because m does not exist.