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(Section 7)
A thorough knowledge of graphing linear and non-linear equations is extremely helpful for a visual understanding of many basic calculus concepts. Therefore, it is very important that any student beginning a study of calculus have a firm comprehension of the following graphing ideas.

Straight Lines

Straight lines:
y = 5 a horizontal line through y = 5 (a linear function of x)
x = -7 a vertical line through x = 5 (a linear function of y, but not a linear function of x)
graph

 

 

Point-Slope Form
y - y1 = m(x - x1) a line through (x1,y1) with a slope of m
y - 3 = 6(x + 2) a line through (-2,3) with a slope of 6 
graph
Slope-Intercept Form
y = mx + b a line through (0,b) with a slope of m
y = -2x + 8 a line through (0,8) with a slope of -2
graph

Taylor Form

EQUATION a line through point with a slope of m
equation a line through (3,2) with a slope of number.
This form is useful when entering an equation in a graphing calculator since all the equation must be entered to the right of the equal sign.
Parallel Lines
Two lines are parallel if and only if they have equal slopes but different y-intercepts:
graph
graph
Perpendicular Lines
Two lines are perpendicular if and only if they have slopes whose product is -1.
graph
graph
Warning: Sometimes these lines may not appear to be perpendicular. Many graphing calculator windows are not square, but instead rectangular. If windows are not square, the lines with negative reciprocal slopes will not appear perpendicular.

 

Periodically in Calculus you must recall the Distance Formula, the midpoint of a segment, and the angle of elevation.  

Distance Between Two Points

The distance, d, between (x1,y1) and (x2,y2) is equation
 

The distance between (3,4) and (-5,8) is 


 
equation

equation
 

equation
 

or equation

equation

graph

Midpoint of a Line Segment

If   and  are the end points of a line segment, then the midpoint of the line segment is
 

graph

 
Let (-2,4) and (-8,6) be endpoints of a line segment. To find the midpoint of the segment you can average the ordered pairs for the endpoints: 

midpoint

midpoint
 

(-3,5)
 

graph

Angle of Elevation
 

If a non-vertical line l intersects the x-axis at an angle t and has a slope of m then tan t = m.

 
The line y = (x + 2) + 2, which is the Taylor form for y = x + 4, has a slope of 1 and intersects the x-axis at , therefore . So the angle of elevation for this line is
graph

Check Your Understanding: (Remember to write down your answers.)
 

 

Check Your Understanding: (Remember to write down your answers.)
Are these lines perpendicular, parallel or neither?
1. y = 3x + 8 and 3x + y = 10
2. 4x + 3y = 12 and 3x - 4y = 24
3. x - y = 12 and x - y = 15
4. Write the equation, in point-slope form, of a line through (-2,1) with a slope of 2.
5. Write the equation, in slope-intercept form, of a line parallel to y = 3x - 8 that passes through (2,1). Enter this equation in your calculator and check to see that (2,1) is a solution to your equation.
6. Write the equation, in point-slope form, of a line perpendicular to x + 2y = 8 that passes through the origin. Graph both lines on a square window to check to see that the lines are visually perpendicular.
7. Write the equation, in Taylor form, of a line parallel to 3x - 2y = 7 that passes through (3,1). Enter this equation in your calculator and check to see that (3,1) is a solution to your equation.

8.  Find the distance between (-2,6) and (4,8)

9.  Find the midpoint of the line segment whose endpoints are (-2,6) and (4,8).

10.  Find the angle of elevation for the line equation

 

Table of Contents

Section 6

Answer Section

Section 8