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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.9144 & x_2 = 2.3731 \end{matrix} $$

(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) = -2x^4+6x^3-2x^2-4x+4 } $, so:

$$ \text{Y inercept} = p(0) = 4 $$

STEP 3:Find the end behavior

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

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

The graph starts in the lower-left corner.

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

The graph ends in the lower-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = -8x^3+18x^2-4x-4 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.3527 & x_2 = 0.7763 & x_3 = 1.8263 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.3527 } \Rightarrow p\left(-0.3527\right) = \color{orangered}{ 4.8678 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.7763 } \Rightarrow p\left(0.7763\right) = \color{orangered}{ 1.7701 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.8263 } \Rightarrow p\left(1.8263\right) = \color{orangered}{ 4.323 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.3527, 4.8678 \right) & \left( 0.7763, 1.7701 \right) & \left( 1.8263, 4.323 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -24x^2+36x-4 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.3792 & x_2 = 0.1208 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.3792 } \Rightarrow p\left(1.3792\right) = \color{orangered}{ 3.183 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.1208 } \Rightarrow p\left(0.1208\right) = \color{orangered}{ 3.4976 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.3792, 3.183 \right) & \left( 0.1208, 3.4976 \right)\end{matrix} $$

Graphing Polynomials