Reflect on the concept of polynomial and rational functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest polynomial and rational function you can imagine? In your day to day, is there any occurring fact that can be interpreted as polynomial and rational functions? What strategy are you using to get the graph of polynomial and rational functions?
Answer:
For polynomials, I needed to understand terms like coefficients, exponents, degree, leading term, and constant term. And for rational functions, I needed to grasp the concepts of numerator, denominator, domain, and vertical asymptotes. These concepts help me understand and work with polynomial and rational functions.
Now, let’s talk about the simplest examples of polynomial and rational functions! The simplest polynomial function I can imagine is a constant function, like f(x) = 5. It has a degree of 0 and represents a horizontal line that never changes. On the other hand, the simplest rational function I can think of is f(x) = 1/x. It has a numerator of 1 and a denominator of x, and it represents a hyperbola that approaches the x and y axes as x approaches positive or negative infinity.
In our day-to-day lives, there are several occurrences that can be interpreted as polynomial and rational functions. For example, if you’re driving a car and you measure the distance you’ve traveled over time, you can model that relationship with a polynomial function. The distance you’ve traveled can be represented by a polynomial function of time, where the coefficients and exponents define the relationship between the two variables. The height of an object thrown into the air can also be modeled by a polynomial function.
Similarly, rational functions can be applied to real-life scenarios. For instance, if you’re calculating the average speed of a moving object, you can express it as a rational function. The average speed is given by the ratio of the total distance traveled to the total time taken. The ratio of the number of boys to the number of girls in a classroom can also be represented by a rational function.
Now, let’s talk about strategies for graphing polynomial and rational functions. To get the graph of a polynomial function, I usually rely on a few key steps. First, I identify the degree of the polynomial and determine its end behavior based on the leading term. Then, I find the x-intercepts by setting the polynomial equal to zero and solving for x. Next, I locate the y-intercept by evaluating the polynomial at x = 0. Finally, I plot additional points if needed and connect them to create a smooth curve.
When it comes to graphing rational functions, I follow a similar approach. I start by finding the domain of the function by identifying any values of x that would make the denominator equal to zero (these are the vertical asymptotes). Then, I determine the horizontal asymptotes by comparing the degrees of the numerator and denominator. After that, I locate any x-intercepts by setting the numerator equal to zero and solving for x. Finally, I plot the points and connect them to form the graph.
Of course, graphing can sometimes be a bit tricky, especially for more complex functions. In those cases, I might use graphing calculators or software to help me visualize the function accurately.