Sketch the graph of the polynomial function $$ p(x) = 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20} $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20} = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0.9021 & x_2 = -1.0736 \end{matrix} $$

(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) = 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20} } $, so:

$$ \text{Y inercept} = p(0) = 6 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20} \right) = \lim_{x \to -\infty} 2x^3 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20} \right) = \lim_{x \to \infty} 2x^3 = \color{blue}{ \infty } $$

The graph ends in the upper-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = -80x^{19}-33x^{10}+6x^5+5x^4+6x^2-2x-7 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -0.9103 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.9103 } \Rightarrow p\left(-0.9103\right) = \color{orangered}{ 10.4351 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( -0.9103, 10.4351 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -1520x^{18}-330x^9+30x^4+20x^3+12x-2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.1585 & x_2 = -0.6926 & x_3 = 0.7238 & x_4 = -0.8287 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.1585 } \Rightarrow p\left(0.1585\right) = \color{orangered}{ 4.8737 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.6926 } \Rightarrow p\left(-0.6926\right) = \color{orangered}{ 9.7051 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.7238 } \Rightarrow p\left(0.7238\right) = \color{orangered}{ 1.4185 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8287 } \Rightarrow p\left(-0.8287\right) = \color{orangered}{ 10.1955 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.1585, 4.8737 \right) & \left( -0.6926, 9.7051 \right) & \left( 0.7238, 1.4185 \right) & \left( -0.8287, 10.1955 \right)\end{matrix} $$

Graphing Polynomials