REVIEW OF LINES AND SLOPES OF LINES

For success in PreCalculus and Calculus, you must have an extremely high comfort level working with lines and slopes of lines.

This web exercise offers an in-a-nutshell review of essential concepts.
It might be all that you need!

For a thorough review, however, go through the following web exercises in order.
Be sure to click-click-click the exercises in each section to check your understanding.

SLOPE OF LINES essential concepts
  • ‘slope’ is a number that measures the ‘steepness’ of a line;
    it is conventionally denoted by the variable $\,m\,$
  • Let $\,(x_1,y_1)\,$ and $\,(x_2,y_2)\,$ be two points on a non-vertical line. Then:
    $$m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1}$$
  • ‘rise’ denotes vertical travel:
    • up is positive
    • down is negative
  • ‘run’ denotes horizontal travel:
    • to the right is positive
    • to the left is negative
  • to find the slope of any non-vertical line:
    • choose any two points
    • travel from one to the other, going vertically first (the rise)
      and horizontally second (the run)
    • take the rise, and divide by the run
  • horizontal lines have slope zero ($m = 0$)
  • vertical lines have no slope (the slope does not exist, due to division by zero)
  • two nonvertical lines are parallel if and only if they have the same slope
  • perpendicular lines have slopes that are opposite reciprocals;
    for example, lines with slopes $\,3\,$ and $\,-\frac 13\,$ are perpendicular
  • going from left to right:
    • uphill lines have positive slopes
    • downhill lines have negative slopes
    • steep lines have big slopes (numbers far away from zero)
    • gradual lines have small slopes (numbers close to zero)
horizontal line,
zero slope,
equation $\,y = k\,$
vertical line,
no slope,
equation $\,x = k\,$
UPHILL LINES, POSITIVE SLOPES
gradual uphill,
small positive slope
steep uphill,
large positive slope
DOWNHILL LINES, NEGATIVE SLOPES
gradual downhill,
small negative slope
steep downhill,
large negative slope
LINES useful forms
  • standard form, $\,ax + by + c = 0$
    • as long as $\,a\,$ and $\,b\,$ are not both zero, then $\,ax + by + c = 0\,$ graphs as a line
    • when $\,a = 0\,$ there is no $\,x\,$ term, and the line is horizontal;
      horizontal lines can be written in the form $\,y = k$
    • when $\,b = 0\,$ there is no $\,y\,$ term, and the line is vertical;
      vertical lines can be written in the form $\,x = k$
  • slope-intercept form, $\,y = mx + b$
    • every equation of the form $\,y=mx+b\,$ graphs as a non-vertical line
    • the slope of the line is $\,m\,$ (the coefficient of the $\,x\,$ term)
    • the line crosses the $y$-axis at the point $\,(0,b)$
  • point-slope form, $\,y - y_1 = m(x - x_1)$
    • this line has slope $\,m\,$ and passes through the point $\,(x_1,y_1)\,$
LINES key concepts
  • intercepts:
    • to find an $x$-intercept:   set $\,y = 0\,$ and solve for $\,x$
    • to find a $y$-intercept:   set $\,x = 0\,$ and solve for $\,y$
  • For lines, equal changes in the input give rise to equal changes in the output:
    if $\,x\,$ changes by $\,\Delta x\,$ (delta $\,x\,$), then $\,y\,$ changes by $\,m\Delta x\,$.
    That is, the outputs change $\,m\,$ times as fast than the inputs (where $\,m = \text{slope}\,$).
    Example: if a line has slope $\,3\,$, the outputs change $\,3\,$ times as fast as the inputs.

Click here for a great exercise on finding equations of lines, written by one of my users (it opens in a new window).
Click the ‘New line’ button to get started.
Find the $y$-intercept; find another point that lies on the intersections of the dashed grid lines, and determine the slope.
Type your equation at the bottom, and click ‘Graph my equation’ to compare.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Distance Between Points; the Midpoint Formula
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31 32  
AVAILABLE MASTERED IN PROGRESS

(MAX is 32; there are 32 different problem types.)