Sketch the graph of the polynomial function $$ p(x) = 2x^4+14x^3-8x^2-56x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^4+14x^3-8x^2-56x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 2 & x_3 = -2 & x_4 = -7 \end{matrix} $$

(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^4+14x^3-8x^2-56x } $, 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.

$$ \lim_{x \to -\infty} \left( 2x^4+14x^3-8x^2-56x \right) = \lim_{x \to -\infty} 2x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 2x^4+14x^3-8x^2-56x \right) = \lim_{x \to \infty} 2x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 8x^3+42x^2-16x-56 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 1.2075 & x_2 = -1.0776 & x_3 = -5.3799 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.2075 } \Rightarrow p\left(1.2075\right) = \color{orangered}{ -50.3842 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.0776 } \Rightarrow p\left(-1.0776\right) = \color{orangered}{ 36.234 }\\[1 em] \text{for } ~ x & = \color{blue}{ -5.3799 } \Rightarrow p\left(-5.3799\right) = \color{orangered}{ -434.8108 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 1.2075, -50.3842 \right) & \left( -1.0776, 36.234 \right) & \left( -5.3799, -434.8108 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 24x^2+84x-16 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.1811 & x_2 = -3.6811 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.1811 } \Rightarrow p\left(0.1811\right) = \color{orangered}{ -10.319 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.6811 } \Rightarrow p\left(-3.6811\right) = \color{orangered}{ -233.3616 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.1811, -10.319 \right) & \left( -3.6811, -233.3616 \right)\end{matrix} $$

Graphing Polynomials