library

  Function and Relation Library

  LINES:

 
Notes, Discussion, Activities
Vertical Lines,
Horizontal Lines - Constant Functions,
Linear Functions,
Equations of Lines,
Solving Linear Equations Graphically
 
Interactive Spread sheet w/o explaination:
linear.xls - It displays graphs of a linear equation by adjusting slope and y-intercept
 



    Vertical lines are not functions. They are included in this library for completeness.

    So where does the equation x = "some constant" come from?

    Check out the ordered pairs on the line x = -3.7.   Every x value is -3.7.   ANY y, which is used, is matched with the same x value, in this case -3.7, as are all the other y values.

    Vertical lines have no slope since the x value never changes.

    They often occur as vertical asymptotes for other functions. The reciprocal function has as an asymptote the vertical line x = 2. The reciprocal function has as an asymptote the vertical line x = -2. The reciprocal function has as a vertical asymptote the vertical line x = -4.


1.   Write, then check, the equation of the vertical line containing the points (4,1), (4,3), and (4,-6).
Your answer:         
Answer:
2.   Write, then check, the equation of the vertical asymptote of the graph y = 1/(x - 6) + 3.
Your answer:         
Answer:



    Horizontal lines are the graphs of constant functions, those whose values never change no matter what values are acted upon. Any x value produces the same y value as that of all the other x values: the result is constantly the same.

    The slope of a horizontal line is zero.

    They often occur as as the horizontal asymptotes for other functions. The reciprocal function has as an asymptote the horizontal line y = 0. The reciprocal function has as an asymptote the horizontal line y = 0. The reciprocal function has as a horizontal asymptote line y = 3.

    Horizontal lines are polynomials of degree zero: y = a is also where the n, or degree, is zero.

3.   Write, then check, the equation of the horizontal line containing the points (5,1), (4,1).
Your answer:         
Answer:
4.   Write, then check, the equation of the horizontal asymptote of the graph y = 1/(x - 6) + 3.
Your answer:         
Answer:


    Linear equations have expressions which are first degree polynomials. They are not constantly the same. As x changes, the y value changes in some way and it is always the same sort of change. Experiment with this yourself. Enter the value for x. (Enter negatives as "-x" rather than "- x.") Press the keys to produce the values of y.

If x = ,
= ,
= .

    (Just for fun, experiment to see if you can find a value of x which makes the other values equal.)

    The coefficient of the x term is the slope of the line, when the equation is written as an expression equal to y. The slope of y= 3x + 1 is 3 and of y = -4x + 8 is -4.

    The y-intercept is the constant term, when the equation is written as an expression equal to y. The y-intercept of of y= 3x + 1 is 1 and of y = -4x + 8 is 8.

    Linear functions are important.

    Direct variation is described by a linear function: y = kx.

    A linear regression line y = ax + b may be used to describe a relation statistically.

5.   Write, then check, the equation of the a line with a slope of 2 and a y-intercept of 1.
Your answer:         
Answer:
6.   Write, then check, the value of y in the equation y = 2x - 6 when x is 3.
Your answer:         
Answer:





Solving Linear Equations by Graphing
Solve 3x + 1 = -4x + 8
Using the graph of y = 3x + 1
and the graph of y = -4x + 8.
To solve a linear equation by graphing,
· Draw the graph for the expression on the left side,
· Draw the graph for the expression on the right side,
· Determine the x value of the point of intersection of the two lines.
· State the x value, the solution.

    The equation 3x + 1 = -4x + 8 may be solved by graphing y = -4x + 8, y = 3x + 1, and noting that the x-value of their intersection (1,4) is 1. The solution to the equation is 1.

7.   Graphically solve the equation -x + 2 = 1 using graphs
of the lines y = 1 and y = -x + 2.
Your answer:         
Answer:
8.   Graphically solve the euqation -x + 2 = -x - 1 using graphs
of the lines y = -x + 2 and y = -x - 1.
Your answer:         
Answer:







EQUATIONS OF LINES
Slope-Intercept: y = mx + b
Point-Slope: y - y1 = m(x -x1)
Standard: Ax + By = C
General: Ax + By + C = 0

    This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

    You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. YOU MAY NOT MAKE ANY ADDITIONAL COPIES OF THIS PAGE, ITS PICTURE(S), ITS SOUND CLIP(S), OR ITS ANIMATED GIFS WITHOUT PERMISSION.


[MC,i. Home] [Table] [Words] Classes [this semester's schedule w/links] [Good Stuff -- free & valuable resources] [next] [last]
© 2006, Agnes Azzolino
www.mathnstuff.com/math/spoken/here/2class/300/fx/library/lines.htm