Unit V BQ #7

Where does the difference quotient come from?

PC: Kelsea; graph

In the quadratic graph above, there is a specific point labeled (x, f(x)). This specific point creates a tangent line. A Tangent line is a line that only touches the graph once. The second point, (x+h, f(x+h)) creates a secant line. A secant line is a line that touches two points on the graph. Now, you're probably wondering where we go the two points. Well, it's kind of simple even though it looks complicated on the picture right now.

So, let's begin by observing the point (x, f(x)). This point is the original one given to us. Since on the graph, the x value is stated, all we need to do to find the y value is plug it into the function, that is why it is f(x). Now, to the solid point on the x-axis, we call it h since we do not know the exact value of it. Now, how do we find the point on the graph above h? Well, for the x value of it, we know that if we add x+h, it will give us the x-value. Now, to find the y-value, all we do is plug in the x to the function, making it f(x+h). Yay! Now we have the two exact points for our secant line! Now, how do we find the equation for the tangent line of (x+h),f(x+h))? Well, in the picture below, we use the slope equation to help with that. It is worked out below. Now, we can see that this is where the difference quotient comes from! Yay! So, in short, the difference quotient comes from slope equation, that is solved but just didn't use actual numbers.

PC: Kelsea: equation

References:

Mrs. Kirch's videos

Photo's: Kelsea