Function

21 Sunday Apr 2013

Tags

coordinates, equation, function, graph, intercept, line, slope

Definition

A function is an equation between two quantities x and y such that one quantity can be defined in terms of the other quantity.

Example

y = mx + c – eq. 1

Where x and y are two quantities, m is some number, a multiple of x and c is a constant number.

This equation may also be written as:

x = (y – c) / m – eq. 2

Equation eq.1 defines y in yerms of x.

Equation eq.2 defines x in yerms of y.

The function defines a line on a graph. Let m=2 and c=–3, then the equation can be written as

y = 2x – 3

This equation can be plotted on a graph with real values for x and y.

Intercept

When the value of x is set to 0, the line cuts the y-axis of the graph at y = –3. This is called the x-intercept. On the graph, the coordinates are (0, –3).

When the value of y is set to 0, the line cuts the x-axis at 3/2 or 1.5. This is called the y-intercept. On the graph, the coordinates are (3/2, 0).

A straight line can be drawn on the graph with all possible values of x and y between the two intercepts and beyond.

Slope

In the generic function above, m is the called the slope of the line represented by the equation.

The slope can be expressed in terms of x, y and the constant c.

m = (y – c) / x

The slope is a property of the line and is a constant for all values of x and y.

It is calculated by the formula (y1 – y2) / (x1 – x2), where (x1, y1) and (x2, y2) are two points on the line.

For the function y = 2x – 3, the slope obtained is 2 by substituting the intercept points (0, –3) and (3/2, 0) in the slope formula.

Slope of a line may also be expressed in a trigonometric form. The line on the graph subtends an angle with the x-axis. The tangent of this angle is the slope.

If theta is the angle of the line with x-axis, then tan(theta) = opp. Side/adjacent Side.

Therefore, –3/(3/2) , which is 2 below the axis, since the negative sign implies that the angle is below the axis line in the fourth quadrant). See graph below.

Graph

Graph for the function y = 2x – 3 looks as shown below (from WolframAlpha):

Domain

The domain of a function is all values of x.

Range

The range of a function is all values of y corresponding to the values of x.

In the function y = 2x – 3, the domain and range are both real numbers.

Note

Consider an equation of the form ax+by+c=0, where a,b,c are real numbers and x and y are unknown quantities. This equation can be expressed in the function form as shown below:

Dividing the equation by b, it becomes

y = -ax/b-c/b

Where slope m = -ax/b and the constant is -c/b

Slope

22 Wednesday Aug 2012

Posted by Anand Betanabhotla in math

≈ 2 Comments

Tags

coordinates, equation, formula, function, geometry, gradient, intercept, line, slope

The slope of a line measures the deviation of the line from the horizontal. It is usually denoted by the letter m.

The slope is also called the gradient of the line, measured as the angle away from the horizontal axis.

The slope can be expressed in two ways:

From the coordinates of any two points on the line, the slope m is
$m = \frac{\Delta y}{\Delta x} = \frac{change-in-y}{change-in-x} = \frac{y_{1} - y_{2}}{x_{1} - x_{2}} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

In the slope-intercept form as
$y = mx + c$

Note:

The slope of this horizontal line is zero.
(because any two y points on a horizontal line have the same value, their difference is zero)

The slope of a vertical line is undefined.
(because any two points x on a vertical line have the same value, their difference is zero and divide by zero is not defined in mathematics)