# Recent questions in Polynomial graphs

Polynomial graphs

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

Polynomial graphs

### 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.

Polynomial graphs

### 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?

Polynomial graphs

### 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.

Polynomial graphs

### 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 $$\displaystyle{\left[-{20},{20}\right]}.$$

Polynomial graphs

### Graph each polynomial function. $$\displaystyle{f{{\left({x}\right)}}}={4}{x}^{{{4}}}+{7}{x}^{{{2}}}-{2}$$

Polynomial graphs

### Graph each polynomial function. $$f(x)=2x^{3}\ -\ x^{2}\ +\ 2x\ -\ 1$$

Polynomial graphs

### Solve the polynomial inequality graphically. $$\displaystyle{x}^{{{3}}}-{x}^{{{2}}}-{2}{x}\geq{0}$$

Polynomial graphs

### 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?

Polynomial graphs

### 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. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$

Polynomial graphs

### Graph each polynomial function.$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{3}}}+{2}{x}^{{{2}}}-{5}{x}-{6}$$

Polynomial graphs

### Graph each polynomial function. Factor first if the expression is not in factored form. $$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}−{4}\right)}{\left({x}−{1}\right)}$$

Polynomial graphs

### 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?

Polynomial graphs