- 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

- 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.

- y = ax
^{n}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.

- y = ax
^{n}+ b when n = 1 or the equivalent statement y = ax + b or -
y = mx + b or
y - y
_{1}= m(x -x_{1}) 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

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.

**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.

**EQUATIONS OF LINES**- Slope-Intercept: y = mx + b
- Point-Slope:
y - y
_{1}= m(x -x_{1}) - 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.

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