Functions

- Linear
- Piecewise
- Step
- Exponential

Linear functions come in three main forms:

This linear function:

f(x) = mx + b

May be graphed on the x, y plane as this equation:

y = mx + b

This equation is called the slope-intercept form for a line. The graph of this equation is a straight line. The slope of the line is m. The line crosses the y-axis at b. The point where the line crosses the y-axis is called the y-intercept. The x, y coordinates for the y-intercept are (0, b).

How do you graph linear functions?

Example:

To graph a function such as f(x) = y = 2x + 7, we can select a table of values, plot and connect the points. To fill the table, we choose values for x and use the function to get the values of y as shown in the following table.

x | y |

-3.5 | 0 |

-2 | 3 |

-1 | 5 |

0 | 7 |

0.5 | 8 |

1 | 9 |

2 | 11 |

The graph of the data in the above table is:

The value of y when x is zero in the function is called the y-intercept and the value of x when y is zero is called the x-intercept. For more information on intercepts, please refer to intercepts

The above graph is a linear function of the form y = mx + c where m is the slope of the straight line and c is the y-intercept. The slope of a straight line passing through two points (x1, y1) and (x2, y2) is defined as following: m = ( y2 - y1 )/(x2 - x1)

For the above graph, slope = 2 and the y-intercept = 7.

Example: Plot the graph of a linear function 4x - 3y = 12.

Solution:

To plot the graph of a linear function, usually it is sufficient to plot the x and y intercepts of that linear function.

From the above equation, we find that the x-intercept is 3 (i.e. when y = 0) and the y-intercept is -4 (i.e. when x = 0). For convenience, we will also find another point: (x= 6, y=4). Now the graph can be plotted as following:

a more complez form of a piecewise function is as follows

the graph of the above looks like this:

a step function looks like this:

some major exponential rules include:

Example: means which in turn can be written . According to Rule 1, you can get to the answer directly by adding the exponents:

Example: can be written as: .

Example: According to Rule 3, we can go directly to the answer by multiplying the exponents...

.

For a > 0 the value of is always positive, no matter what x is. For every positive the exponential function with base a is one-to-one. It is increasing if a > 1 with a horizontal asymptote at y = 0 on the left, i.e. the graph approaches the x-axis as x->- and decreasing with a horizontal asymptote at y = 0 on the right if a < 1.

For example, if the base is 2 and x = 4, the function value f(4) will equal 16. A corresponding point on the graph of would be (4, 16):

The general behaviour of more complicated functions involving exponentials can often be figured out on the basis of whether the powers involved have positive or negative exponents. The exponent is always negative and becomes even more negative as . So the value of is always less than or equal to1 and the graph has a horizontal asymptote at y = 0 on both sides.

Logarithmic Functions are also related to exponential functions. Logarithmic functions are the inverse of exponential functions. The exponential equation could be written in terms of a logarithmic equation as . The exponential equation can be written as the logarithmic equation .

The graphs are similar too, and since logarithms are nothing more than exponents, you can use the rules of exponents with logarithms.

For example: if (4, 16) is a point on the graph of an exponential function, then (16, 4) would be the corresponding point on the graph of the inverse logarithmic function:

Function Links

To learn more about these functions, visit the following links:

Introduction to Functions

Linear functions

Piecewise functions

Step functions

Exponential functions

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