Question

Christi is doing her math homework. To receive full credit, she must answer this question: What key features are necessary—and how are the features used—to create the sketch of a polynomial function? What is Christi's correct answer, so she receives full credit for the question? Explain in complete sentences.

Answer 1

Christi should know the "zeros" of the polynomial function for sketching the graph of it (where the graph intersects with the x an y-axises, if it does). Also, she needs to know the vertical and horizontal stretches/shrinks for that. Finally, she should know the basic function of the polynomial function such as x^2, 1/x and etc.

Answer 2

Answer:

We are given that, Christi needs to sketch a polynomial function given by $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+.......a_{1}x+a_{0}$.

The few key features necessary to plot a polynomial function are:

1. Zeroes of the polynomial.

In order to plot a polynomial function, we are required to know the x-intercepts i.e. the points at which the graph will cut the x-axis.

They can be obtained by finding the zeroes of a polynomial, which can be found by using,

Fundamental Theorem of Algebra states that 'an n-degree polynomial will have n-zeroes'.

2. Extremum points and the point of inflection.

Now, in order to know the the points where graph of the polynomial function becomes flat, we find the extrema points.

That is, extrema points are the points which makes the slope of the function  zero, which can be obtained by using,

Derivative Test, in which we 'differentiate the function with respect to x and equate it to 0'.

3. End Behavior.

The polynomial function can be sketched easily when we the end behavior of the function, which can be viewed by using,

Leading Coefficient Test, which states the behavior of the polynomial function depending upon the degree and the leading co-efficient of the polynomial.