Sketch the graph of the polynomial function $$ p(x) = x^6-x^5-6x^4-x^2+x+10 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = -1.22 & x_2 = -1.9382 & x_3 = 1.1393 & x_4 = 2.9898 \end{matrix} $$

(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^6-x^5-6x^4-x^2+x+10 } $, so:

$$ \text{Y inercept} = p(0) = 10 $$

STEP 3:Find the end behavior

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

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 6x^5-5x^4-24x^3-2x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.2659 & x_2 = -1.6642 & x_3 = 2.4702 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.2659 } \Rightarrow p\left(0.2659\right) = \color{orangered}{ 10.1642 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.6642 } \Rightarrow p\left(-1.6642\right) = \color{orangered}{ -6.4475 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.4702 } \Rightarrow p\left(2.4702\right) = \color{orangered}{ -81.8111 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.2659, 10.1642 \right) & \left( -1.6642, -6.4475 \right) & \left( 2.4702, -81.8111 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 30x^4-20x^3-72x^2-2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -1.2644 & x_2 = 1.9237 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -1.2644 } \Rightarrow p\left(-1.2644\right) = \color{orangered}{ -0.881 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.9237 } \Rightarrow p\left(1.9237\right) = \color{orangered}{ -49.6069 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -1.2644, -0.881 \right) & \left( 1.9237, -49.6069 \right)\end{matrix} $$

Graphing Polynomials