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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = \dfrac{ 1 }{ 2 } & x_3 = 1.618 & x_4 = -0.618 \end{matrix} $$

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

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

STEP 3:Find the end behavior

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

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 8x^3-9x^2-2x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.2624 & x_2 = -0.3826 & x_3 = 1.2452 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.2624 } \Rightarrow p\left(0.2624\right) = \color{orangered}{ 0.1488 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.3826 } \Rightarrow p\left(-0.3826\right) = \color{orangered}{ -0.3181 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.2452 } \Rightarrow p\left(1.2452\right) = \color{orangered}{ -1.2892 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.2624, 0.1488 \right) & \left( -0.3826, -0.3181 \right) & \left( 1.2452, -1.2892 \right)\end{matrix} $$

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^2-18x-2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.8482 & x_2 = -0.0982 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.8482 } \Rightarrow p\left(0.8482\right) = \color{orangered}{ -0.6668 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.0982 } \Rightarrow p\left(-0.0982\right) = \color{orangered}{ -0.1049 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.8482, -0.6668 \right) & \left( -0.0982, -0.1049 \right)\end{matrix} $$

Graphing Polynomials