Step 1

Step 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 2021-02-26

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 2020-12-03

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 2021-01-06

Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of \(\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}\)

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?

\(\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}\)

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?

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 2021-02-25

In 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. Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?

\(\displaystyle{\arctan{{x}}}\approx{x}-\frac{{x}^{{{3}}}}{{3}}+\frac{{x}^{{{5}}}}{{5}}-\frac{{x}^{{{7}}}}{{7}}\)

where x is in radians. Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?

asked 2021-02-25

Sketch the graphs of two functions that are not polynomial functions.

a) \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}}\)

b) \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}}\)

Explain your reasoning.

a) \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}}\)

b) \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}}\)

Explain your reasoning.

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.

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 2021-01-25

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?