Slope of a Line Lesson 13.2 Pre-AP Geometry

Lesson Focus The purpose of this lesson is to introduce and study the slope of a line.

Slope of a Line Slope is a characteristic of a line that is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline.

Example #1 Find the slope of the line through the points (-4, 1) and (3, 2). or

Practice #1 Find the slope of the line through the points (-2, -1) and (-4, -3).

Trigonometry Connection “If a line with a positive slope makes an acute angle on n  with the x-axis, then the slope of the line is tan n .” Example: Construct a triangle with vertices at A(0, 0), B(5, 0), and C(5, 4). Show that tan  A = slope of segment AC. Use a trig. ratio to find the m  A.

Trigonometry Connection Solution: Note: The tangent of the angle formed by the intersection of a line with a horizontal line gives the slope of the line. That angle is known as the “angle of elevation”.

Types of Slope Positive slope Lines that rise from left to right. The sign of the value of m is positive. Negative slope Lines that fall from left to right. The sign of the value of m is negative. Zero slope Horizontal lines. m = 0 Undefined or Infinite Slope Vertical lines. m = undefined

Positive Slope Lines in slope-intercept form with m > 0 have positive slope. This means for each unit increase in x, there is a corresponding m unit increase in y (i.e. the line rises by m units). Lines with positive slope rise to the right on a graph as shown in the following picture:

Positive Slope Lines with greater slopes rise more steeply. For a one unit increment in x, a line with slope m 1 = 1 rises one unit while a line with slope m 2 = 2 rises two units as depicted:

Negative Slope Lines in slope-intercept form with m < 0 have negative slope. This means for each unit increase in x, there is a corresponding |m| unit decrease in y (i.e. the line falls by |m| units). Lines with negative slope fall to the right on a graph as shown in the following picture:

Negative Slope The steepness of lines with negative slope can also be compared. Specifically, if two lines have negative slope, the line whose slope has greatest magnitude (known as the absolute value) falls more steeply. For a one unit increment in x, a line with slope m 3 = −1 falls one unit while a line with slope m 4 = −2 falls two units as depicted:

Zero Slope As stated above, horizontal lines have slope equal to zero. This does not mean that horizontal lines have no slope. Since m = 0 in the case of horizontal lines, they are symbolically represented by the equation, y = b. Functions represented by horizontal lines are often called constant functions.

Undefined Slope Vertical lines have undefined slope. Since any two points on a vertical line have the same x-coordinate, slope cannot be computed as a finite number according to the slope formula, because division by zero is an undefined operation. Vertical lines are symbolically represented by the equation, x = a where a is the x-intercept. Vertical lines are not functions; they do not pass the vertical line test at the point x = a.

Written Exercises Problem Set 13.2, p.532: # (even); Handout 13-2