# Describe any similarities and differences. Refer to the end behaviour, local maximum and local minimum points, and maximum and minimum points. a) Sketch graphs of y=sin x and y=cos x. b) Compare the graph of a periodic function to the graph of a polynomial function.

Question
Polynomial graphs
Describe any similarities and differences. Refer to the end behaviour, local maximum and local minimum points, and maximum and minimum points.
a) Sketch graphs of $$\displaystyle{y}={\sin{\ }}{x}$$ and $$\displaystyle{y}={\cos{\ }}{x}.$$
b) Compare the graph of a periodic function to the graph of a polynomial function.

2021-02-21
Step 1
a)

Step 2
b) Periodic function: A function with a graph that repeats identically smoothly from left to right. It can be represented by $$\displaystyle{f{{\left({x}\right)}}}={f{{\left({x}\ +\ {p}\right)}}}{>}$$
There is no the end points for the periodic functions and have uncountable number from the turning points (local maximum and local minimum points).
The graphs of polynomial functions are continuous, and smooth graphs.
The turning points and the behavior of polynomial function graphs depend mainly on its degree.

### Relevant Questions

Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$
a) Identify the parameters a, k, d, and c in the polynomial function $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{4}}}$$. b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function. c) Describe two possible orders in which the transformations can be applied to the graph of $$\displaystyle{y}={x}^{{{4}}}$$ to produce the graph of $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. d) Sketch graphs of the base function and the transformed function on the same set of axes.
Investigate the change in the graph of a sinusoidal function of the form $$\displaystyle{\quad\text{and}\quad}={\sin{{x}}}{\quad\text{or}\quad}{\quad\text{and}\quad}={\cos{{x}}}$$ when multiplied by a polynomial function. Use a graphing calculator to sketch the graphs of and $$\displaystyle={2}{x},{\quad\text{and}\quad}=-{2}{x},{\quad\text{and}\quad}={2}{x}{\cos{{x}}}$$ on the same coordinate plane, on the interval $$\displaystyle{\left[-{20},{20}\right]}.$$
For the following exercise, for each polynomial $$\displaystyle{f{{\left({x}\right)}}}=\ {\frac{{{1}}}{{{2}}}}{x}^{{{2}}}\ -\ {1}$$: a) find the degree, b) find the zeros, if any, c) find the y-intercept(s), if any, d) use the leading coefficient to determine the graph’s end behavior, e) determine algebraically whether the polynomial is even, odd, or neither.
Describe the similarities between a) the lines $$\displaystyle{y}={x}{\quad\text{and}\quad}{y}=-{x}$$ and the graphs of other odd-degree polynomial functions b) the parabolas $$\displaystyle{y}={x}^{{{2}}}{\quad\text{and}\quad}{y}=-{x}^{{{2}}}$$ and the graphs of other even-degree polynomial functions
a) Identify the parameters a, b, h, and k in the polynomial $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left({x}+{3}\right)}^{{{3}}}-{2}$$ Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{3}}}$$.
b) State the domain and range of the transformed function.
c) Sketch graphs of the base function and the transformed function on the same set of axes.
Find the quadratic polynomial $$\displaystyle{g{{\left({x}\right)}}}-{a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c}\ \text{which best fits the function}\ {f{{\left({x}\right)}}}={e}^{{{x}}}\ \text{at}\ {x}={0},\ \text{in the sense that}\ {g{{\left({0}\right)}}}={f{{\left({0}\right)}}},\ \text{and}\ {g}'{\left({0}\right)}={f}'{\left({0}\right)},\ \text{and}\ {g}{''}{\left({0}\right)}={f}{''}{\left({0}\right)}.$$ Using a computer or calculator, sketch graphs of f and g on the same axes. What do you notice?
$$g(x)=-x^{2}+2x+6$$
$$f(x)=8x-\frac{9}{x}$$
$$\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}$$