Polynomial Equations and Graphs: Discover the Relationship

High School
Algebra
Algebra II
Polynomial graphs

Explain what you know about the roots of polynomial functions. How can you determine all of the roots? Give examples, including sketches of graphs.

vestirme4 2021-05-23

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?

Brittney Lord 2021-05-21

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at $x = m i n u s; 3$ and $x = 2$ and a root of multiplicity 1 at $x = m i n u s; 2$ . y-intercept at (0, 4).

ringearV 2021-05-19

Solve the polynomial inequality graphically.
$- x^{3} - 3 x^{2} - 9 x + 4 < 0$

Lipossig 2021-05-17

For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. $f (x) = 3 x - x^{3}$

naivlingr 2021-05-13

Find a formula for the function described. A fourth-degree polynomial whose graph is symmetric about the y-axis, has a y-intercept of 0, and global maxima at (1, 2) and (-1, 2).

Haven 2021-05-09

Graph the polynomial by transforming an appropriate graph of the form $y = x^{n}$ . Show clearly all x- and y-intercepts. $P (x) = - 3 {(x + 2)}^{5} + 96$

Reeves 2021-05-08

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at $x = 3$ and $x = 1$ , and a root of multiplicity 1 at $x = - 3$ . y-intercept at (0, 9)

Tyra 2021-05-05

Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $P (x) = a (x - α) (x - β) (x - γ), a \neq 0$
Type 2: Two real zeros, one repeated: $P (x) = a (x - α) 2 (x - β), a \neq 0$
Type 3: One real zero repeated three times: $P (x) = a (x - α) 3, a \neq 0$

Type 4: One real and two imaginary zeros: $P (x) = (x - α) (a x 2 + b x + c), △ = b 2 - 4 a c < 0, a \neq 0$
Experiment with the graphs of Type 4 cubics. What is the geometrical significance of $α α$ and the quadratic factor which has imaginary zeros?

Tammy Todd 2021-05-04

For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.
$f (x) = - 3 x^{2} + 6 x$

iohanetc 2021-03-08

For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at $x = - 3$ and triple zero $a t x = 0$ . Passes through the point (1, 32).

foass77W 2021-03-07

Write an equation for the polynomial graph:

$y (x) = ?$

slaggingV 2021-02-25

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

Haven 2021-02-25

Write an equation for the polynomial graph:

y(x)=?

Chaya Galloway 2021-02-20

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. $y = x^{3} - 3 x^{2}, [- 2, 5] b y [- 10, 10]$

Emily-Jane Bray 2021-02-14

Find the cubic polynomial whose graph passes through the points $(- 1, - 1), (0, 1), (1, 3), (4, - 1) .$

arenceabigns 2021-01-07

Describe the similarities between a) the lines $y = x and y = - x$ and the graphs of other odd-degree polynomial functions b) the parabolas $y = x^{2} and y = - x^{2}$ and the graphs of other even-degree polynomial functions

sanuluy 2021-01-06

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

DofotheroU 2021-01-05

Let $S_{N} (x) = \frac{4}{π} \sum_{n = 1}^{N} \frac{1 - {(- 1)}^{n}}{n^{3}} \sin (n x) .$
Construct graphs of $S_{N} (x) and x (π - x), f or 0 \leq x \leq π, f or N = 2 and t h e n N = 10 .$
This will give some sense of the correctness of Fourier’s claim that this polynomial could be exactly represented by the infinite series
$\frac{4}{π} \sum_{n = 1}^{\infty} \frac{1 - {(- 1)}^{n}}{n^{3}} \sin (n x) o n [0, π] .$

tabita57i 2020-12-30

What can you say about the graphs of polynomial functions with an even degree compared to the graphs of polynomial functions with an odd degree? Use graphs from the Polynomial Functions Investigation (and maybe some others) to justify your response.

If you are looking for polynomial graphs examples, do not forget about word problems as well because these will help you explain your questions that are dealing with polynomial problems. Always start with the composition of the graph and only then continue with the transformation of variables or the use of an equation like in the case with polynomial equations and graphs. See some examples below and approach them the same way with a detailed analysis. If you need to start with your polynomial graph to equation homework, these samples with answers will be essential.

Algebra II

Polynomial Graphs Examples: See the Power of Polynomials in Action