STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^4+13x^3-49x^2-24x+108 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 2 & x_2 = -9 & x_3 = -\dfrac{ 3 }{ 2 } & x_4 = 2 \end{matrix} $$

(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+13x^3-49x^2-24x+108 } $, so:

$$ \text{Y inercept} = p(0) = 108 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 2x^4+13x^3-49x^2-24x+108 \right) = \lim_{x \to -\infty} 2x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 2x^4+13x^3-49x^2-24x+108 \right) = \lim_{x \to \infty} 2x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 8x^3+39x^2-98x-24 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 2 & x_2 = -0.2256 & x_3 = -6.6494 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2 } \Rightarrow p\left(2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.2256 } \Rightarrow p\left(-0.2256\right) = \color{orangered}{ 110.7764 }\\[1 em] \text{for } ~ x & = \color{blue}{ -6.6494 } \Rightarrow p\left(-6.6494\right) = \color{orangered}{ -1811.0787 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 2, 0 \right) & \left( -0.2256, 110.7764 \right) & \left( -6.6494, -1811.0787 \right)\end{matrix} $$

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+78x-98 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.9681 & x_2 = -4.2181 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.9681 } \Rightarrow p\left(0.9681\right) = \color{orangered}{ 52.3969 }\\[1 em] \text{for } ~ x & = \color{blue}{ -4.2181 } \Rightarrow p\left(-4.2181\right) = \color{orangered}{ -1005.0853 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.9681, 52.3969 \right) & \left( -4.2181, -1005.0853 \right)\end{matrix} $$

Graphing Polynomials