# Algebra II: Polynomial graphs questions and answers

Recent questions in Polynomial graphs
Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x = −3, x = −2 and x = 1. y-intercept at (0, 12).

Polynomial graphs

### Graph each polynomial function. Estimate the x-coordinate at which the relative maxima and relative minima occur. State the domain and range for each function. f(x)=−x^3+4x^2−2x−1

Polynomial graphs

### Find a nonzero polymonial function. The given zeros are: a) 0, 1, 9 b) -3, -1, 0, 1, 3

Polynomial graphs

### Simplify the difference quotients between $$\displaystyle{f{{\left({x}+{h}\right)}}}-\frac{{f{{\left({x}\right)}}}}{{h}}{\quad\text{and}\quad}{f{{\left({x}\right)}}}-\frac{{f{{\left({a}\right)}}}}{{{x}-{a}}}$$, if $$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}^{{2}}-{7}}}$$

Polynomial graphs

### Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. $$\displaystyle{f{{\left({x}\right)}}}={x}+{e}^{{-{x}}},{a}={0}$$

Polynomial graphs

### $$\displaystyle{f{{\left({x}\right)}}}={100}{\left({1.25}\right)}^{{x}}$$ Is it true or false, the growth factor is 25%? If not, what is and what is 25%?

Polynomial graphs

### $$\displaystyle{\left(\frac{{2}}{{7}},-{1}\right)}{\quad\text{and}\quad}{9}+\frac{{1}}{{{3}{i}}}$$ are zeros. Find a polynomial function with real coefficients.

Polynomial graphs

### $$\displaystyle{y}={x}^{{3}}-{2}{x}^{{2}}-{x}+{2}$$ and $$\displaystyle{y}=-{x}^{{2}}+{5}{x}+{2}$$ To determine:a) Graph of both functions b) At how many points do the both graphs appear to intersect. c) Find coordinates of all intersection points

Polynomial graphs

### $$\displaystyle{f{{\left({x}\right)}}}={x}^{{5}}+{2}{x}^{{4}}+{3}{x}^{{3}}−{2}{x}^{{6}}−{9}{x}^{{2}}−{6}{x}+{4}$$ Is it true or false the degree of this polynomial function is 5? If not, why? What is the degree?

Polynomial graphs

### Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of polynomial functions have no vertical asymptotes.

Polynomial graphs

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

Polynomial graphs

### Investigate the change in the graph of a sinusoidal function of the form $$\displaystyle{\quad\text{and}\quad}={\sin{{x}}}$$ or $$\displaystyle{\quad\text{and}\quad}={\cos{{x}}}$$ when multiplied by a polynomial function. Use a graphing calculator to sketch the graphs of $$\displaystyle{\quad\text{and}\quad}={x}^{{{2}}},{\quad\text{and}\quad}=−{x}^{{{2}}}$$ and $$\displaystyle{\quad\text{and}\quad}={x}^{{{2}}}{\sin{{x}}}$$ on the same coordinate plane, on the interval [-20, 20].

Polynomial graphs

### Graph the polynomial by transforming an appropriate graph of the form $$\displaystyle{y}={x}^{{{n}}}$$. Show clearly all x- and y-intercepts. $$\displaystyle{P}{\left({x}\right)}={81}-{\left({x}-{3}\right)}^{{{4}}}$$

Polynomial graphs

### The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are. $$\begin{array} &x&-2&-1&0&1&2\\f(x)&0.8&0.2&0.1&0.005&0.025\\g(x)&80&40&20&10&2 \end{array}$$

Polynomial graphs

### Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{\left({x}−β\right)}{\left({x}−γ\right)},{a}≠{0}$$ Type 2: Two real zeros, one repeated: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{2}{\left({x}−β\right)},{a}≠{0}$$ Type 3: One real zero repeated three times: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{3},{a}≠{0}$$ Type 4: One real and two imaginary zeros: $$\displaystyle{P}{\left({x}\right)}={\left({x}−α\right)}{\left({a}{x}{2}+{b}{x}+{c}\right)},Δ={b}{2}−{4}{a}{c}{<}{0},{a}≠{0}$$ Experiment with the graphs of Type 1 cubics. Clearly state the effect of changing both the size and sign of a. What is the geometrical significance of $$\alpha, \beta, and\ \gamma ? \alpha,\beta,and\ \gamma$$?

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Double zero at $$x = 1$$, and triple zero at $$x = 3$$. Passes through the point (2, 15).

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 $$x2>4$$ 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 $$x2 >4$$ 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

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Root of multiplicity 2 at $$x = 4,$$ and roots of multiplicity 1 at $$x = 1$$ and $$x = minus$$;2. y-intercept at (0, minus;3).

Polynomial graphs