Sketch the graph of the polynomial function $$ p(x) = x^4-3x^3-36x^2+68x+240 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^4-3x^3-36x^2+68x+240 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 4 & x_2 = 6 & x_3 = -2 & x_4 = -5 \end{matrix} $$

(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^4-3x^3-36x^2+68x+240 } $, so:

$$ \text{Y inercept} = p(0) = 240 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^4-3x^3-36x^2+68x+240 \right) = \lim_{x \to -\infty} x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^4-3x^3-36x^2+68x+240 \right) = \lim_{x \to \infty} x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 4x^3-9x^2-72x+68 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.8851 & x_2 = -3.753 & x_3 = 5.118 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.8851 } \Rightarrow p\left(0.8851\right) = \color{orangered}{ 270.5179 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.753 } \Rightarrow p\left(-3.753\right) = \color{orangered}{ -165.2937 }\\[1 em] \text{for } ~ x & = \color{blue}{ 5.118 } \Rightarrow p\left(5.118\right) = \color{orangered}{ -71.0171 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.8851, 270.5179 \right) & \left( -3.753, -165.2937 \right) & \left( 5.118, -71.0171 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 12x^2-18x-72 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 3.3117 & x_2 = -1.8117 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 3.3117 } \Rightarrow p\left(3.3117\right) = \color{orangered}{ 81.6872 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.8117 } \Rightarrow p\left(-1.8117\right) = \color{orangered}{ 27.2503 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 3.3117, 81.6872 \right) & \left( -1.8117, 27.2503 \right)\end{matrix} $$

Graphing Polynomials