Sketch the graph of the polynomial function $$ p(x) = x^4+10x^3-411x^2-360x+13500 $$

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+10x^3-411x^2-360x+13500 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 6 & x_2 = 15 & x_3 = -6 & x_4 = -25 \end{matrix} $$

(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+10x^3-411x^2-360x+13500 } $, so:

$$ \text{Y inercept} = p(0) = 13500 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^4+10x^3-411x^2-360x+13500 \right) = \lim_{x \to -\infty} x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^4+10x^3-411x^2-360x+13500 \right) = \lim_{x \to \infty} x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 4x^3+30x^2-822x-360 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.4316 & x_2 = 11.3332 & x_3 = -18.4017 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.4316 } \Rightarrow p\left(-0.4316\right) = \color{orangered}{ 13578.0462 }\\[1 em] \text{for } ~ x & = \color{blue}{ 11.3332 } \Rightarrow p\left(11.3332\right) = \color{orangered}{ -12315.6543 }\\[1 em] \text{for } ~ x & = \color{blue}{ -18.4017 } \Rightarrow p\left(-18.4017\right) = \color{orangered}{ -66696.3294 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.4316, 13578.0462 \right) & \left( 11.3332, -12315.6543 \right) & \left( -18.4017, -66696.3294 \right)\end{matrix} $$

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+60x-822 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 6.1458 & x_2 = -11.1458 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 6.1458 } \Rightarrow p\left(6.1458\right) = \color{orangered}{ -488.379 }\\[1 em] \text{for } ~ x & = \color{blue}{ -11.1458 } \Rightarrow p\left(-11.1458\right) = \color{orangered}{ -31959.121 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 6.1458, -488.379 \right) & \left( -11.1458, -31959.121 \right)\end{matrix} $$

Graphing Polynomials