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## PowerPoint Slideshow about 'Linear Functions' - grietje

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### Graphs of Linear Functions

Definition

A function f is linear if its domain is a set of numbers and it can be expressed in the form

where m and b are constants and x denotes an arbitrary element of the domain of f.

Definition

- If x1 and x2 are distinct members of the domain of f, the change in f from x1 to x2 is f(x2) – f(x1). The rate of change of f over the interval from x1 to x2 is

Notation

- Let Dx = x2 – x1 denote the change in x. Let Df =f(x2) – f(x1) denote the change in f.
- The rate of change is the ratio

Exercise

- For real numbers x, let . Find the change in f from x1 = 1 to x2 = 4.
- Find the rate of change of f over the interval from 0 to 3 .
- Find a general formula for the rate of change over the interval from x1 to x2 for any x1 and x2.

A Characterization of Linear Functions

A function from the real numbers to the real numbers is linear if and only if its rate of change is the same for all intervals. If so, the rate of change is the constant m in the formula

Straight Lines

in the plane determine one and only one straight line

Equation: x = c.

Point-Slope Form

Suppose it is known that a line passes through the point with coordinates and that it has slope m. Then the equation of the line is

Slope Intercept Form

- y = f(x) = mx + b
- m = rate of change of f = slope of the line = tangent of angle between the x-axis and the line
- b = f(0) = y-intercept of the line

The Symmetric Form

- Slope-intercept and point-slope forms cannot handle vertical lines in the xy plane.
- Symmetric form does not select one variable as the independent variable and the other as the dependent variable. c, d, and e are constants.

Exercise

The graph of a linear function is the line whose equation is

What is the rate of change of f? What are f(0) and f(-2)?

General Form of a Linear System of Two Equations in Two Unknowns

Equations in Symmetric

Form of Two Straight Lines

Three Possibilities for Solutions Unknowns

- The lines are not parallel and intersect in one and only one point. That is, there is one and only one solution of the system.
- The lines are distinct but parallel and do not intersect. There are no solutions.
- The equations represent the same straight line. There are infinitely many solutions, one for each point on the line.

The Coefficient Matrix Unknowns

The Determinant of the Coefficient Matrix Unknowns

The number

Relationship of the Determinant to the Question of Solutions Unknowns

The linear system has a unique solution if and only if the determinant is different from zero.

Exercise Unknowns

- Solve
- Answer: x=3/7, y=2/7

Inverses of Linear Functions Unknowns

Example Unknowns

Given y, solve for x:

Example (continued) Unknowns

The equation

defines x as a linear function of y. This function is called the inverse of the original function. We write

Equivalence Unknowns

The two equations

and

are equivalent. One is satisfied by a pair (x,y) if and only if the other is.

General Expression for the Inverse Function Unknowns

- If f (x) = mx + b and m≠0, then
- Note: The slope of the inverse function is the reciprocal of the slope of the original function.

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