Sketch the graph of the polynomial function $$ p(x) = x^5-10x^3+12x+2 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^5-10x^3+12x+2 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.1708 & x_2 = -1.0695 & x_3 = 1.2726 & x_4 = 2.9171 & x_5 = -2.9493 \end{matrix} $$

(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^5-10x^3+12x+2 } $, so:

$$ \text{Y inercept} = p(0) = 2 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^5-10x^3+12x+2 \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5-10x^3+12x+2 \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4-30x^2+12 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.6565 & x_2 = -0.6565 & x_3 = -2.3599 & x_4 = 2.3599 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.6565 } \Rightarrow p\left(0.6565\right) = \color{orangered}{ 7.1705 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.6565 } \Rightarrow p\left(-0.6565\right) = \color{orangered}{ -3.1705 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.3599 } \Rightarrow p\left(-2.3599\right) = \color{orangered}{ 31.9143 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.3599 } \Rightarrow p\left(2.3599\right) = \color{orangered}{ -27.9143 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.6565, 7.1705 \right) & \left( -0.6565, -3.1705 \right) & \left( -2.3599, 31.9143 \right) & \left( 2.3599, -27.9143 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 20x^3-60x $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0 & x_2 = \sqrt{ 3 } & x_3 = -\sqrt{ 3 } \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 2 }\\[1 em] \text{for } ~ x & = \color{blue}{ \sqrt{ 3 } } \Rightarrow p\left(\sqrt{ 3 }\right) = \color{orangered}{ -13.5885 }\\[1 em] \text{for } ~ x & = \color{blue}{ -\sqrt{ 3 } } \Rightarrow p\left(-\sqrt{ 3 }\right) = \color{orangered}{ 17.5885 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0, 2 \right) & \left( \sqrt{ 3 }, -13.5885 \right) & \left( -\sqrt{ 3 }, 17.5885 \right)\end{matrix} $$

Graphing Polynomials