## LINES:

Notes, Discussion, Activities
Vertical Lines,
Horizontal Lines - Constant Functions,
Linear Functions,
Equations of Lines,
Solving Linear Equations Graphically

linear.xls - It displays graphs of a linear equation by adjusting slope and y-intercept

 Visit the Interactive Sketch Pad Material On Lines line   -- slide points A and B and generate equation of the line in 4 forms.

VERTICAL LINES
• x = a, where a is a constant
• ex. x = -3.7
• NOT a function
• no slope or infinite slope [See Slope in Sound & Picture]

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).
2.   Write, then check, the equation of the vertical asymptote of the graph y = 1/(x - 6) + 3.

HORIZONTAL LINES - CONSTANT FUNCTIONS
• y = axn when n = 0 or the equivalent statement y = a
• ex. y = 3.7
• opposite, additive inverse: y = - ax
• multiplicative inverse, reciprocal: y = a/x
• slope: a
• inverse function: It has no inverse. It fails the horizontal line test because each y value is the same.

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).
4.   Write, then check, the equation of the horizontal asymptote of the graph y = 1/(x - 6) + 3.

Linear Equations
• y = axn + b when n = 1 or the equivalent statement y = ax + b or
• y = mx + b or y - y1 = m(x -x1) or Ax + By + c = 0 or Ax + By = C
• ex. y = 3x + 1, ex. y = -4x + 8
• opposite, additive inverse: y = - ax - b
• multiplicative inverse, reciprocal: y = 1/(ax + b)
• slope: a
• inverse function: y = (x - b)/a or y = ((1/a)x - b/a

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.
6.   Write, then check, the value of y in the equation y = 2x - 6 when x is 3.

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.
8.   Graphically solve the euqation -x + 2 = -x - 1 using graphs
of the lines y = -x + 2 and y = -x - 1.

EQUATIONS OF LINES
Slope-Intercept: y = mx + b
Point-Slope: y - y1 = m(x -x1)
Point-Slope: form used in Calc I to approximate the value of a function
y = m(x -x1) - y1 or
f(x) = f'(x)(x -x1) - f(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).

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