Sketch the graph of the polynomial function $$ p(x) = x^4+2x^3-6x^2+1 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^4+2x^3-6x^2+1 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0.4522 & x_2 = -0.3885 & x_3 = 1.5677 & x_4 = -3.6314 \end{matrix} $$

(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+2x^3-6x^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.

$$ \lim_{x \to -\infty} \left( x^4+2x^3-6x^2+1 \right) = \lim_{x \to -\infty} x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^4+2x^3-6x^2+1 \right) = \lim_{x \to \infty} x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 4x^3+6x^2-12x $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = 1.1375 & x_3 = -2.6375 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 1 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.1375 } \Rightarrow p\left(1.1375\right) = \color{orangered}{ -2.1456 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.6375 } \Rightarrow p\left(-2.6375\right) = \color{orangered}{ -29.0419 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0, 1 \right) & \left( 1.1375, -2.1456 \right) & \left( -2.6375, -29.0419 \right)\end{matrix} $$

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+12x-12 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.618 & x_2 = -1.618 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.618 } \Rightarrow p\left(0.618\right) = \color{orangered}{ -0.6738 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.618 } \Rightarrow p\left(-1.618\right) = \color{orangered}{ -16.3262 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.618, -0.6738 \right) & \left( -1.618, -16.3262 \right)\end{matrix} $$

Graphing Polynomials