Slopes of Equations and Lines Honors Geometry Chapter 2 Nancy Powell, 2007

Objectives: Calculate and interpret the slope of a line Graph lines given a point and the slope Use the point-slope form of a line Find the equation of a line given two points Write the equation of a line in slope-intercept form

slope m Let and be two distinct points with. The slope m of the non-vertical line is defined by the formula undefined If, then is a vertical line and the slope m is undefined (since this results in division by 0). NOTE:

y x Slope can be thought of as the ratio of the vertical change ( ) to the horizontal change ( ), often termed RISE RUN RISE over RUN

y x +RISE +RUN +RISE +RUN RISING Because this line rises from left to right with a +RISE and a +RUN, we’ll call this a RISING line. x

y x -RISE +RUN – RISE +RUN FALLING Because this line falls from left to right or has a – RISE and a +RUN, we’ll call this a FALLING line.

y x UNDEFINED + or - RISE Zero RUN) If, then is zero and the slope is UNDEFINED. ( + or - RISE and Zero RUN) Plotting the two points results in the graph of a VERTICAL line with the equation. x1x1

y x ZERO Zero RISE + or - RUN) If, then is zero and the slope is ZERO. ( Zero RISE and + or - RUN) Plotting the two points results in the graph of a HoRIZ0ntal line with the equation.

Example: Find the slope of the line joining the points (3,8) and (-1,2).

Example: Draw the graph of the line passing through (1,4) with a slope of -3/2. Step 1: Plot the given point. Step 2: Use the slope to find another point on the line (vertical change =, horizontal change = ). y x ( 1,4 ) 2 -3 (3,1) Find another point on this line. ( ____, ____ ) - 3 2

Example: Draw the graph of the equation x = 2. y x x = 2 What do you know about the slope of this line? The slope is undefined.

Theorem: Point-Slope Form of an Equation of a Line An equation of a non-vertical line of slope m that passes through the point (x 1, y 1 ) is:

Example: Find an equation of a line with slope -2 passing through (-1,5).

A HoRIZ0ntal line is given by an equation of the form y = b, where (0,b) is the y- intercept. Slope = Example: Graph the line y=4. y x y = 4 0

The equation of a line L is in general form with it is written as where A, B, and C are three integers and A and B are not both 0. The equation of a line in slope-intercept form is written as y = mx + b where m is the slope of the line and (0,b) is the y-intercept.

Example: Find the slope m and y-intercept (0,b) of the graph of the line 3x - 2y + 6 = 0. Solve for y in terms of x and find the slope and the y-intercept! So, (0,3) is the y-intercept

Objectives (part 2): Define parallel, perpendicular, and oblique lines Find equations of parallel Lines Find equations of perpendicular Line Determine whether lines are parallel, perpendicular or oblique

Definition: Parallel Lines l m n parallel Two distinct non-vertical lines are parallel if and only if they same plane 1.are in the same plane, same slope 2.have the same slope and different y-intercepts 3.have different y-intercepts.

Find the equation of the line parallel to y = -3x + 5 passing through (1,7). Since parallel lines have the same slope, the slope of the parallel line must also be equal to -3. Isn’t this Slope- intercept form? x1 = 1 and y1 = 7

Definition: Perpendicular Lines perpendicular Two lines are said to be perpendicular if they intersect at a right angle. Two non-vertical lines are perpendicular if slopes are opposite reciprocalsTheir slopes are opposite reciprocals of each other like product of their slopes is -1The product of their slopes is -1.

Example: Find the equation of the line perpendicular to y = -3x + 10 passing through (1,5). Slope of perpendicular line: l m n Isn’t this Slope- intercept form?

Definition: Oblique Lines o Oblique lines are lines that intersect but are not perpendicular to each other. o Oblique lines do not form right angles. o This means that the slopes of oblique lines are not the same and they are not opposite reciprocals. l m n

Which of the following pairs of slopes are slopes of o Oblique lines? o Parallel lines? o Perpendicular lines? How do you know? a. 2/3 and 3/2 c. 2 and e and g. ¾ and 0.75 Slopes and Oblique, Parallel, and Perpendicular Lines a. 2/3 and 3/2 b. -2/3 and -3/2 d.5/4 and 2/3 f.7/4 and 3/7 h. 0 and undefined b. -2/3 and -3/2 d.5 /4 and 2/3 e and f.7 /4 and 3/7 g. ¾ and 0.75 c. 2 and h. 0 and undefined