## Polynomial and Rational Functions

Completely analyse functions to degree 5 Use synthetic division to factor functions

## Characteristics of Functions

### Rates of Change

What we are interested in here is how fast a function is increasing or decreasing at any particular point. If we are dealing with a linear function, then we are just looking for the slope of the line. We just need to re-arrange the equation into the slope intercept form and read the slope. What do we do about non-linear equations? You can't read the slope from the equation, and in fact, the slope changes from point to point. Now think back to one of the ways that we can find the equation of a line given two points $(x_1,y_1),(x_2,y_2)$ the slope of the line can be found with the formula $S = \frac{y_2-y_1}{x_2-x_1}$, or rise over run, and then find the intercept. So if we pick two points on the graph of the equation, we can find the rate of change using rise over run, or even find the equation of the line through these two points. If we pick two points, say (2,f(2)) and (3,f(3)), then we can find the average rate of change between those points. Now, when we say average here, we're talking about the physics notion of average, rather than the statistical average. Consider if I take a trip from Cambridge to Ottawa, with a distance of 527km, and I complete it in 6 hours, my average speed is 87km/h. Which I get from dividing the distance by the time. My instantaneous speed is what my speedometer reads at any given point in the journey, which can range from mumble-mumble when I'm cruising down the 401 to 0 when I'm stopped for a coffee. If I were to plot my distance travelled versus time, then my instantaneous speed at any given time would be the slope of that line at that point. I can estimate that instantaneous speed at, say, time = 3hours, by finding the slope between two points on the graph that are very close to t=3. There are a few method to choose these points one is the preceding/following method, where for one point you use t=3, and for the other choose t=3.01 or t=2.99, for example. The second method is the centered interval method where you would choose your two points to be t=2.99 and t=3.01. The final method is to figure out the difference quotient. This is a generalized method of calculating the following method, by simplifying the formula ${f(a+h)-f(a)}\over{h}$ and then subbing in the value of x and a suitably small value for h. This is only worth the bother if you need to find the slope at several points on the same graph.

### Functions as Models

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