Sketch the graph of the polynomial function $$ p(x) = x^4+3x^3-43x^2-87x+126 $$

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^3-43x^2-87x+126 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1 & x_2 = 6 & x_3 = -3 & x_4 = -7 \end{matrix} $$

(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) = x^4+3x^3-43x^2-87x+126 } $, so:

$$ \text{Y inercept} = p(0) = 126 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^4+3x^3-43x^2-87x+126 \right) = \lim_{x \to -\infty} x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^4+3x^3-43x^2-87x+126 \right) = \lim_{x \to \infty} x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 4x^3+9x^2-86x-87 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.9566 & x_2 = 4.1653 & x_3 = -5.4587 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.9566 } \Rightarrow p\left(-0.9566\right) = \color{orangered}{ 168.0869 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.1653 } \Rightarrow p\left(4.1653\right) = \color{orangered}{ -464.6059 }\\[1 em] \text{for } ~ x & = \color{blue}{ -5.4587 } \Rightarrow p\left(-5.4587\right) = \color{orangered}{ -280.4615 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.9566, 168.0869 \right) & \left( 4.1653, -464.6059 \right) & \left( -5.4587, -280.4615 \right)\end{matrix} $$

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-86 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 2.0301 & x_2 = -3.5301 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2.0301 } \Rightarrow p\left(2.0301\right) = \color{orangered}{ -185.7569 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.5301 } \Rightarrow p\left(-3.5301\right) = \color{orangered}{ -79.4167 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 2.0301, -185.7569 \right) & \left( -3.5301, -79.4167 \right)\end{matrix} $$

Graphing Polynomials