# Graph the polynomial function. f(x)=−x^{4}+3x^{3}−x+1

Question
Polynomial graphs
Graph the polynomial function. $$\displaystyle{f{{\left({x}\right)}}}=−{x}^{{{4}}}+{3}{x}^{{{3}}}−{x}+{1}$$

2021-02-26
Step 1
$$\displaystyle{f{{\left({x}\right)}}}=-{x}^{{{4}}}+{3}{x}^{{{3}}}-{x}+{1}$$
The leading coefficient is -1 (negative) and the degree is 4 (even)
Therefore $$\displaystyle{f{{\left({x}\right)}}}\rightarrow-\infty,{a}{s}\ {x}\rightarrow\infty$$
and $$\displaystyle{f{{\left({x}\right)}}}\rightarrow-\infty,{a}{s}\ {x}\rightarrow-\infty$$
Step 2

### Relevant Questions

For each polynomial function, one zero is given. Find all rational zeros and factor the polynomial. Then graph the function. f(x)=3x^{3}+x^{2}-10x-8ZSK, zero:2
Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. $$\displaystyle{y}={x}^{{{3}}}-{3}{x}^{{{2}}},{\left[-{2},{5}\right]}{b}{y}{\left[-{10},{10}\right]}$$
Graph each polynomial function. Factor first if the expression is not in factored form. PSKf(x)=x^{3}\ +\ 3x^{2}\ -\ 13x\ -\ 15 PSZ
Graph each polynomial function. Factor first if the expression is not in factored form.
$$\displaystyle{f{{\left({x}\right)}}}={\left({3}{x}-{1}\right)}{\left({x}+{2}\right)}^{{{2}}}$$
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. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$
Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. $$\displaystyle{\left({2},\ {1}\right)},\ {\left({3},\ {1}\right)},\ {\left({4},\ −{3}\right)},\ {\left({5},\ {0}\right)}$$.
Graph each polynomial function. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{4}{x}^{{{3}}}-{5}{x}^{{{2}}}+{14}{x}-{15}$$
Solve the polynomial inequality graphically. $$\displaystyle{3}{x}^{{{4}}}-{5}{x}^{{{3}}}-{12}{x}^{{{2}}}+{12}{x}+{16}$$
Graph each polynomial function. Factor first if the expression is not in factored form. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}$$