Sketch the graph of the polynomial function $$ p(x) = 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20}+x^{19}-55x^{13} $$

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}+x^{19}-55x^{13} = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0.7502 & x_2 = -1.4268 \end{matrix} $$

(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}+x^{19}-55x^{13} } $, 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.

$$ \lim_{x \to -\infty} \left( 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20}+x^{19}-55x^{13} \right) = \lim_{x \to -\infty} 2x^3 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 2x^3-x^2-7x+6+x^6+x^5-3x^{11}-4x^{20}+x^{19}-55x^{13} \right) = \lim_{x \to \infty} 2x^3 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = -80x^{19}+19x^{18}-715x^{12}-33x^{10}+6x^5+5x^4+6x^2-2x-7 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -1.3408 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -1.3408 } \Rightarrow p\left(-1.3408\right) = \color{orangered}{ 901.8065 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( -1.3408, 901.8065 \right)\end{matrix} $$

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}+342x^{17}-8580x^{11}-330x^9+30x^4+20x^3+12x-2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.1585 & x_2 = -0.5297 & x_3 = 0.5345 & x_4 = -1.255 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.1585 } \Rightarrow p\left(0.1585\right) = \color{orangered}{ 4.8737 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.5297 } \Rightarrow p\left(-0.5297\right) = \color{orangered}{ 9.1275 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5345 } \Rightarrow p\left(0.5345\right) = \color{orangered}{ 2.326 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.255 } \Rightarrow p\left(-1.255\right) = \color{orangered}{ 649.7632 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.1585, 4.8737 \right) & \left( -0.5297, 9.1275 \right) & \left( 0.5345, 2.326 \right) & \left( -1.255, 649.7632 \right)\end{matrix} $$

Graphing Polynomials