# Describe the similarities between a) the lines y = x and y = -x and the graphs of other odd-degree polynomial functions b) the parabolas y=x^{2} and y=-x^{2} and the graphs of other even-degree polynomial functions

Question
Polynomial graphs
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

2021-01-08
Step 1
The graphs of the Odd Degree Polynomial Functions will depend mainly on the Leading Coefficient.
If it is Positive the graph will expand feom quadrant 3 to quadrant 1 i.e. like to the graph of $$\displaystyle{y}={x}$$
On the other hand, If it is Negative the graph will expand from quadrant 2 to quadrant 4 i.e. like to the graph of $$\displaystyle{y}=-{x}$$
Step 2
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Step 3
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Step 4 The Graphs of the Even Degree Polynomial Functions will depend mainly on the Leading Coefficient.
If it is Positive the graph will expand from quadrant 2 to quadrant 1 i.e. like to the graph of PSKy=x^{2}.
On the other hand, If it is Negative the graph will expand from quadrant 3 to quadrant 4 i.e. like to the graph of $$\displaystyle{y}=-{x}^{{{2}}}$$
Step 5

Step 6

a) Graph of odd-degree polynomial functions and the lines $$\displaystyle{y}={x}{\quad\text{and}\quad}{y}=-{x}$$
b) Graph of even-degree polynomial functions and the lines $$\displaystyle{y}={x}^{{{2}}}{\quad\text{and}\quad}{y}=-{x}^{{{2}}}$$

### Relevant Questions

What can you say about the graphs of polynomial functions with an even degree compared to the graphs of polynomial functions with an odd degree? Use graphs from the Polynomial Functions Investigation (and maybe some others) to justify your response.
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.
Make rough sketches of the graphs of each of the following polynomial functions. Be sure to label the x- and y- intercepts. $$\displaystyle{a}{)}{y}={x}{\left({2}{x}+{5}\right)}{\left({2}{x}-{7}\right)}$$ $$\displaystyle{b}{)}{y}={\left({15}-{2}{x}\right)}^{{{2}}}{\left({x}+{3}\right)}$$
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.
Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of $$\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}$$
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, 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 $$\displaystyle{y}={x}^{{{3}}}$$.
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.
Sketch graphs of each of the following polynomial functions. Be sure to label the x- and they-intercepts of each graph. $$\displaystyle{a}.{y}={x}{\left({2}{x}+{3}\right)}{\left({2}{x}-{5}\right)}{b}.{y}={\left({11}-{2}{x}\right)}^{{{2}}}{\left({x}-{2}\right)}$$
(a) find the Maclaurin polynomial $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ for f(x), (b) complete the following $$\displaystyle{x}:-{0.75},-{0.50},-{0.25},{0},{0.25},{0.50},{0.75}{f}{\quad\text{or}\quad}{f{{\left({x}\right)}}}$$ and $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$, and (c) sketch the graphs of f(x) and $$\displaystyle{P}_{{{3}}}{\left({x}\right)}$$ on the same set of coordinate axes. $$\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}$$