Sketch the graph of the polynomial function $$ p(x) = x^{11}+x^{10}+2x^9+3x^8+x^7+x^6+8x^5+2x^4+x^3-x^2-8x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^{11}+x^{10}+2x^9+3x^8+x^7+x^6+8x^5+2x^4+x^3-x^2-8x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 0.8493 & x_3 = -1.0358 \end{matrix} $$

(you can use the step-by-step polynomial equation solver to see a detailed explanation of how to solve the equation)

STEP 2:Find the y-intercepts

To find the y-intercepts, substitute $ x = 0 $ into $ \color{blue}{ p(x) = x^{11}+x^{10}+2x^9+3x^8+x^7+x^6+8x^5+2x^4+x^3-x^2-8x } $, so:

$$ \text{Y inercept} = p(0) = 0 $$

STEP 3:Find the end behavior

The end behavior of a polynomial is the same as the end behavior of a leading term.

$$ \lim_{x \to -\infty} \left( x^{11}+x^{10}+2x^9+3x^8+x^7+x^6+8x^5+2x^4+x^3-x^2-8x \right) = \lim_{x \to -\infty} x^{11} = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^{11}+x^{10}+2x^9+3x^8+x^7+x^6+8x^5+2x^4+x^3-x^2-8x \right) = \lim_{x \to \infty} x^{11} = \color{blue}{ \infty } $$

The graph ends in the upper-right corner.

STEP 4:Find the turning points

To determine the turning points, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = 11x^{10}+10x^9+18x^8+24x^7+7x^6+6x^5+40x^4+8x^3+3x^2-2x-8 $$

The x coordinate of the turning points are located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.5887 & x_2 = -0.687 \end{matrix} $$

(cleck here to see a explanation of how to solve the equation)

To find the y coordinates, substitute the above values into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.5887 } \Rightarrow p\left(0.5887\right) = \color{orangered}{ -3.9119 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.687 } \Rightarrow p\left(-0.687\right) = \color{orangered}{ 4.042 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.5887, -3.9119 \right) & \left( -0.687, 4.042 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 110x^9+90x^8+144x^7+168x^6+42x^5+30x^4+160x^3+24x^2+6x-2 $.

The zero of second derivative is

$$ \begin{matrix}x = 0.15 \end{matrix} $$

Substitute the x value into $ p(x) $ to get y coordinates

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.15 } \Rightarrow p\left(0.15\right) = \color{orangered}{ -1.2173 }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( 0.15, -1.2173 \right)\end{matrix} $$

Graphing Polynomials