Sketch the graph of the polynomial function $$ p(x) = 2x^5+14x^4-12x^3-144x^2 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^5+14x^4-12x^3-144x^2 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 3 & x_3 = -4 & x_4 = -6 \end{matrix} $$

(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^5+14x^4-12x^3-144x^2 } $, 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.

$$ \lim_{x \to -\infty} \left( 2x^5+14x^4-12x^3-144x^2 \right) = \lim_{x \to -\infty} 2x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 2x^5+14x^4-12x^3-144x^2 \right) = \lim_{x \to \infty} 2x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 10x^4+56x^3-36x^2-288x $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = 2.1707 & x_3 = -2.5332 & x_4 = -5.2374 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.1707 } \Rightarrow p\left(2.1707\right) = \color{orangered}{ -394.0352 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.5332 } \Rightarrow p\left(-2.5332\right) = \color{orangered}{ -361.1146 }\\[1 em] \text{for } ~ x & = \color{blue}{ -5.2374 } \Rightarrow p\left(-5.2374\right) = \color{orangered}{ 426.4406 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0, 0 \right) & \left( 2.1707, -394.0352 \right) & \left( -2.5332, -361.1146 \right) & \left( -5.2374, 426.4406 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 40x^3+168x^2-72x-288 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.3171 & x_2 = -1.2946 & x_3 = -4.2225 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.3171 } \Rightarrow p\left(1.3171\right) = \color{orangered}{ -227.1641 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.2946 } \Rightarrow p\left(-1.2946\right) = \color{orangered}{ -183.2621 }\\[1 em] \text{for } ~ x & = \color{blue}{ -4.2225 } \Rightarrow p\left(-4.2225\right) = \color{orangered}{ 101.8398 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.3171, -227.1641 \right) & \left( -1.2946, -183.2621 \right) & \left( -4.2225, 101.8398 \right)\end{matrix} $$

Graphing Polynomials