Sketch the graph of the polynomial function $$ p(x) = 2372x^4-447x^3-7x^2+14439x+12 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2372x^4-447x^3-7x^2+14439x+12 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.0008 & x_2 = -1.7654 \end{matrix} $$

(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) = 2372x^4-447x^3-7x^2+14439x+12 } $, so:

$$ \text{Y inercept} = p(0) = 12 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 2372x^4-447x^3-7x^2+14439x+12 \right) = \lim_{x \to -\infty} 2372x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 2372x^4-447x^3-7x^2+14439x+12 \right) = \lim_{x \to \infty} 2372x^4 = \color{blue}{ \infty } $$

The graph ends in the upper-right corner.

STEP 4:Find the turning points

To determine the turning point, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = 9488x^3-1341x^2-14x+14439 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -1.1054 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -1.1054 } \Rightarrow p\left(-1.1054\right) = \color{orangered}{ -11812.1187 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( -1.1054, -11812.1187 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 28464x^2-2682x-14 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.0992 & x_2 = -0.005 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.0992 } \Rightarrow p\left(0.0992\right) = \color{orangered}{ 1443.8319 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.005 } \Rightarrow p\left(-0.005\right) = \color{orangered}{ -59.603 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.0992, 1443.8319 \right) & \left( -0.005, -59.603 \right)\end{matrix} $$

Graphing Polynomials