## Linear Functions can be thought of as being written in the form y = a(x-k)+h, for parameters a, k, and h. In this exploration, we will take a look at what effect each of these three parameters have on the graph of a linear function. To begin, let's hold a equal to 1, k equal to 0, and look at varying values of h. Below you will see the graph of y = x + h, for 9 cases: when h = -4, -3, -2, -1, 0, 1, 2, 3, and 4.

## The line at the far right (the second pink one) is the graph for the case when h is equal to -4, and as you move to the left, each line shows the case for -3, -2, -1, 0, 1, 2, 3, and 4, respectively. We can see that the parameter h seems to effect the x-intercept of our function (in these cases, the x-intercept is equal to additive inverse of the parameter h). Now let's take a look at a graph that shows changes in the parameter h as a movie.

## We see the constant movement of the linear function as the parameter h shifts from -4 all the way up to +4. It is helpful to see it in this manner so as to avoid the misconception that the function only exists at integer values of h and hops from each of the lines shown in the first picture to the next. The movement we see here reminds us that the parameter h can take on any real number value, and non-integer values do not change the expected behavior of our function.

## Now let's take a look at what happens when we change the parameter k. Taking the same 9 functions from before, let's see what happens to our graph if we let k be equal to 2. See the graph below for a visual.

## We can see that the only thing that has changed here is that the x-intercept is no longer equal to -h. All of our x-intercepts have shifted over to the right by two (or the parameter k). This makes sense algebraically, since we can simplify the function

## y = (x-2)+(-4) to read y = x+(-6), so we would expect the x-intercept to take place at 6 instead of 4. Let's take a look at this function as we constantly change h. See the movie below.

## Again, seeing the function in constant motion helps us remember that it takes on functional values when the parameter his a non-integer as well as when it is a whole number. Finally, let's take a look at changes in the parameter a when k is held constant at 2 and h is held constant at zero. See the graph below for a visual of what the function looks like when a is equal to -4, -3, -2, -1, 0, 1, 2, 3, and 4.

## We see here that the parameter a effects the overall slope or steepness of our function. When a is positive, we see that the graph increases in value from right to left, and when a is negative the graph decreases in value from left to right. A special case occurs when a is equal to zero. The function takes on a constant value of zero in this case and forms a line across the x-axis. We can also look at changes in this parameter as it moves constantly from -4 to +4.

## Again, this fluid movement and change in the parameter allows us to dynamically see that when k and h are held constant, the function rotates around the fixed point of (2,0), only changing in steepness or slope as a changes.

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