Sketch the graph of the polynomial function $$ p(x) = x^3+6x-6x^2+1 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^3+6x-6x^2+1 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.1451 & x_2 = 1.476 & x_3 = 4.6691 \end{matrix} $$

(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^3+6x-6x^2+1 } $, so:

$$ \text{Y inercept} = p(0) = 1 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^3+6x-6x^2+1 \right) = \lim_{x \to -\infty} x^3 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^3+6x-6x^2+1 \right) = \lim_{x \to \infty} x^3 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 3x^2-12x+6 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 3.4142 & x_2 = 0.5858 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 3.4142 } \Rightarrow p\left(3.4142\right) = \color{orangered}{ -8.6569 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5858 } \Rightarrow p\left(0.5858\right) = \color{orangered}{ 2.6569 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 3.4142, -8.6569 \right) & \left( 0.5858, 2.6569 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 6x-12 $.

The zero of second derivative is

$$ \begin{matrix}x = 2 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2 } \Rightarrow p\left(2\right) = \color{orangered}{ -3 }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( 2, -3 \right)\end{matrix} $$

Graphing Polynomials