Step 1

The expression is given as

\(\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}\)

Step 2

The expression is given as

\(\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}\)

Step 2

Question

asked 2020-10-23

Make rough sketches of the graphs of each of the following polynomial functions. Be sure to label the x- and y- intercepts.
\(\displaystyle{a}{)}{y}={x}{\left({2}{x}+{5}\right)}{\left({2}{x}-{7}\right)}\)
\(\displaystyle{b}{)}{y}={\left({15}-{2}{x}\right)}^{{{2}}}{\left({x}+{3}\right)}\)

asked 2021-03-18

Sketch graphs of each of the following polynomial functions. Be sure to label the x- and they-intercepts of each graph. \(\displaystyle{a}.{y}={x}{\left({2}{x}+{3}\right)}{\left({2}{x}-{5}\right)}{b}.{y}={\left({11}-{2}{x}\right)}^{{{2}}}{\left({x}-{2}\right)}\)

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 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 2020-12-30

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.

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 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 2020-12-15

(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. \(\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}\)

asked 2021-01-25

Using calculus, it can be shown that the secant function can be approximated by the polynomial \(\displaystyle{\sec{{x}}}\approx{1}+{\frac{{{x}^{{{2}}}}}{{{2}!}}}+{\frac{{{5}{x}^{{{4}}}}}{{{4}!}}}\) where x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

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.

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.