# Polynomial graphs with equation

Recent questions in Polynomial graphs
Polynomial graphs

### Solve the polynomial inequality graphically. $$\displaystyle−{x}^{{3}}−{3}{x}^{{2}}−{9}{x}+{4}{<}{0}$$

Polynomial graphs

### The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning points at x = -2, 0, and 3. a) Draw the graphs of two different polynomial functions that have these three characteristics. b) What additional characteristics would ensure that only one graph could be drawn?

Polynomial graphs

### If the result of a polynomial division is $$\displaystyle{2}{x}^{{{3}}}−{4}{x}^{{{2}}}−{5}{x}+{23}{x}+{2}$$ , make at least three definitive statements

Polynomial graphs

### Do you agree with the contention that the functions $$f(x) = x + 2$$ and $$g(x)=\displaystyle{\frac{{{x}^{{{2}}}-{4}}}{{{x}-{2}}}}$$ are the same in every respect. Provide evidence to support your position.

Polynomial graphs

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

Polynomial graphs

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

Polynomial graphs

### Graph the polynomial by transforming an appropriate graph of the form $$\displaystyle{y}={x}^{{{n}}}$$. Show clearly all x- and y-intercepts. $$\displaystyle{P}{\left({x}\right)}=-{x}^{{{3}}}+{64}$$

Polynomial graphs

### Copy and complete the anticipation guide in your notes. StatementThe quadratic formula can only be used when solving a quadratic equation.Cubic equations always have three real roots.The graph of a cubic function always passes through all four quadrants.The graphs of all polynomial functions must pass through at least two quadrants.The expression $$x^2>4$$ is only true if x>2.If you know the instantaneous rates of change for a function at x = 2 and x = 3, you can predict fairly well what the function looks like in between. Agree Disagree Justification Statement The quadratic formula can only be used when solving a quadratic equation. Cubic equations always have three real roots. The graph of a cubic function always passes through all four quadrants. The graphs of all polynomial functions must pass through at least two quadrants. The expression $$x^2>4$$ is only true if x>2. If you know the instantaneous rates of change for a function at x = 2 and x = 3, you can predict fairly well what the function looks like in between.

Polynomial graphs

### Graph each polynomial function. $$\displaystyle{f{{\left({x}\right)}}}={4}{x}^{{{5}}}-{8}{x}^{{{4}}}-{x}+{2}$$

Polynomial graphs

### Graph each polynomial function. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{3}}}+{3}{x}^{{{2}}}-{4}{x}-{2}$$

Polynomial graphs

### Graph the polynomial by transforming an appropriate graph of the form $$\displaystyle{y}={x}^{{{n}}}$$. Show clearly all x- and y-intercepts.$$\displaystyle{P}{\left({x}\right)}={2}{\left({x}+{1}\right)}^{{{4}}}-{32}$$

Polynomial graphs

### Graph the polynomial function. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{5}}}+{x}^{{{2}}}-{4}$$

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

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