Function and Relation Library

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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

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). Your answer: Answer:: Your answer:; Answer:

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

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). Your answer: Answer:: Your answer:; Answer:

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

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

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:: Your answer:; Answer:

6. Write, then check, the value of y in the equation y = 2x - 6 when x is 3. Your answer: Answer:: 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:: 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:: 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); 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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