A linear function is a function in the form f(x) = mx + b.   The graph of a linear function is a line with slope m and y-intercept b.

For example what is the slope of the line h(x) = 2x + 5?   Calculus help is possible with us.
In this case, it is 2 represented by the x coefficient. 

What is the y-intercept?
This is found by plugging in 0 for x and solving for y. 
h(0) = 2(0) + 5
h(0) = 0 + 5
h(0) = 5

To graph a linear function using the slope and y-intercept, first plot the point corresponding to the y-intercept.  What is this point?

It is simply the point calculated above at x= 0 and y=5, therefore (0,5).

Plot an additional point by moving 1 unit to the right of the y-intercept and then up or down the number of units indicated by the slope.  What is the resulting point?

For this problem we need to use our linear equation to solve for 1 unit to the right of the y-intercept.  This would make x=1. 
h(1) = 2(1) + 5
h(1) = 2 + 5
h(1) = 7
Therefore our resulting point is (1,7).

Now we will graph our linear equation.




Here is where most people require the most calculus help. 
The domain of a function h is the largest set of real numbers for which the value of h(x) is a real number.  The range of a function h is the set of h(x)-values of the function that are images of the x-values in the domain.

What is the domain of h(x) = 2x + 5?
(-infinite, infinite)

In the case of h(x) = 2x + 5 the range of x values goes forever, even though on the graph it appears to end at 0 and 3.  You can plug any value into x and you will receive a real y value. 

What is the range of h(x) = 2x + 5?
(-infinite, infinite)

Same as the domain, even though it appears to end at 5 and 11, the y values go infinite in both directions, both below to x axis and above it.  You can plug in any value of x to get a value for y.

The slope is also known as the average rate of change.  So what would be the rate of change for this graph?
2, it is the rate in which adding 1 to x changes the y value. 

Since our slope is positive, we can assume that the domain is forever increasing and is decreasing over its domain if the slope is negative.  The rate of change is constant if the slope is zero.
So for a slope of 2, it is both positive and increasing.

Thus, for the linear function h(x) = 2x + 5, the slope is 2, the y-intercept is 5, the average rate of change is 2, and the function is increasing over its domain.  The graph has a domain and range of
(-infinite, infinite).

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