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.

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
asked 2021-02-20
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.

Answers (1)

2021-02-21
Step 1
a)
image
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.
0

Relevant Questions

asked 2020-12-03
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. \(\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}\)
asked 2021-02-26
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.
asked 2021-02-22
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 \(\displaystyle{\left[-{20},{20}\right]}.\)
asked 2020-11-01
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.
asked 2021-01-07
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
asked 2021-02-25
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.
asked 2021-01-30
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?
asked 2021-06-05
Find the absolute maximum value and the absolute minimum value, if any, of the function.
\(g(x)=-x^{2}+2x+6\)
asked 2021-05-11
Find the absolute maximum value and the absolute minimum value, if any, of the function.
\(f(x)=8x-\frac{9}{x}\)
asked 2020-11-05
Using calculus, it can be shown that the arctangent function can be approximated by the polynomial
\(\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}\)
where x is in radians.
a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
b) Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part (a). How does the accuracy of the approximation change when an additional term is added?
...