Sketch the graph of the polynomial function $$ p(x) = x^8-4x^6+2x^4+4x^2-2 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^8-4x^6+2x^4+4x^2-2 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0.6945 & x_2 = -0.6945 & x_3 = 1.2319 & x_4 = -1.7123 & x_5 = -1.2319 & x_6 = 1.7123 \end{matrix} $$

(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^8-4x^6+2x^4+4x^2-2 } $, so:

$$ \text{Y inercept} = p(0) = -2 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^8-4x^6+2x^4+4x^2-2 \right) = \lim_{x \to -\infty} x^8 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^8-4x^6+2x^4+4x^2-2 \right) = \lim_{x \to \infty} x^8 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 8x^7-24x^5+8x^3+8x $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = 1 & x_3 = -1 & x_4 = -1.5538 & x_5 = 1.5538 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ -2 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1 } \Rightarrow p\left(1\right) = \color{orangered}{ 1 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1 } \Rightarrow p\left(-1\right) = \color{orangered}{ 1 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.5538 } \Rightarrow p\left(-1.5538\right) = \color{orangered}{ -3 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.5538 } \Rightarrow p\left(1.5538\right) = \color{orangered}{ -3 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0, -2 \right) & \left( 1, 1 \right) & \left( -1, 1 \right) & \left( -1.5538, -3 \right) & \left( 1.5538, -3 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 56x^6-120x^4+24x^2+8 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -0.6648 & x_2 = 0.6648 & x_3 = 1.3687 & x_4 = -1.3687 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.6648 } \Rightarrow p\left(-0.6648\right) = \color{orangered}{ -0.1485 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.6648 } \Rightarrow p\left(0.6648\right) = \color{orangered}{ -0.1485 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.3687 } \Rightarrow p\left(1.3687\right) = \color{orangered}{ -1.4694 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.3687 } \Rightarrow p\left(-1.3687\right) = \color{orangered}{ -1.4694 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -0.6648, -0.1485 \right) & \left( 0.6648, -0.1485 \right) & \left( 1.3687, -1.4694 \right) & \left( -1.3687, -1.4694 \right)\end{matrix} $$

Graphing Polynomials