# 1.1 - MATH 1081  Wednesday, January 12   ...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 1081  Wednesday, January 12    Section 1.1    Slopes and Equations of Lines    Homework #1 (due 1/24):   Section 1.1 #36, 64, 70, 72                  Section 2.1 #36, 44, 46, 64 What is the rate of change of the graph at each of the labeled  points?    In order to formalize the notion of the rate of change of a curve  at a given point, we are going to first make the question a  simpler one and go back to something with which we are  already familiar.    What is the rate of change of the line shown here?    Slope of a Line    The rate of change of a line, also called the slope of the line, is  defined as the vertical change (the “rise”) over the horizontal  change (the “run”) as one travels along the line.    That is, if ( x1 , y1 )  and ( x2 , y2 ) are different points on the line,  then the slope is    Change in y Δy y2 − y1 ,  m= = = Change in x Δx x2 − x1   where  x1 ≠ x2 .    What kind of line can have two different points with the same  x ‐coordinate?  What is the slope of such a line?      What kind of line can have two different points with the same  y ‐coordinate?  What is the slope of such a line?      The phrase “no slope” should not be used, because it is  ambiguous.  Example:  Determine the slope of the line through the given  points.    a.  (1, 4 )  and ( 4, −5 )     b.  ( −2, 3)  and ( 5, 5 )     c.  ( 8, 9 )  and ( 8,1)      d.  (10, 3)  and ( −15, 3)     Equations of Lines    In order to determine the equation of a line, we need to know  the slope and the coordinates of one point.  Of course, as we  just reviewed, we can find the slope of a line given any two  different points.    One point of particular interest is the point where the line  crosses the  y ‐axis, as every non‐vertical line must.  The  coordinates of this point will have the form ( 0, b )  for some  number  b  (which may also be 0).    Equation          y = mx + b                  y − y1 = m ( x − x1 )      x = k       y = k                             Description  Slope‐intercept form:    slope  m  and  y ‐intercept  b   Point‐slope form:  slope  m  and passing  through the point ( x1 , y1 )   Vertical line:    slope undefined  Horizontal line:    slope 0                                                          Example:  Determine the equation of the line with the   indicated properties.  Also, determine if the line  passes through the origin (the point ( 0, 0 ) ).    a.    Slope  −2  and  y ‐intercept 3      b. Slope  5  passing through the point (1, 2 )       c.    Horizontal line through the point ( −1, −2 )     4⎞ ⎛ d. Through the points ( −3,1)  and  ⎜ 4, − ⎟   ⎝ 3⎠ Parallel and Perpendicular Lines    Two different lines are parallel, if either they are both vertical  or if they have the same slope.    Two lines are perpendicular, if they intersect at a 90‐degree  angle.  When the neither line is vertical, the product of their  slopes isequal to  −1, or equivalently, the slope of one line is the  negative reciprocal of the slope of the other.     Example:  Determine the equation of the line with the   indicated properties.    a.   Passing through the point ( −2, 6 )  and parallel to     the line  2 x − 3y = 5 .      b. Passing through the origin and perpendicular to  the line through the points ( −3, 0 ) and (1, 5 ) .    Example:  The sales of a small company were \$27,000 in its  second year of operation and \$63,000 in its fifth  year of operation.  Let  y  represent sales in the  x th  year of operation, and assume that the data can be  approximated with a linear equation.    a.    Find an equation for the sales  y  in year  x .    b. According to this model, what will the sales be in      the eighth year of operation?    c.    According to this model, in which year will sales      reach \$250,000?      Tomorrow in recitation: Workshop #1    Next time:  Section 2.1   We will begin using clickers.  ...
View Full Document

## This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

Ask a homework question - tutors are online