Sketch the graph of the polynomial function $$ p(x) = 3x^4-11x^3-6x^2+8x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 3x^4-11x^3-6x^2+8x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 4 & x_3 = -1 & x_4 = \dfrac{ 2 }{ 3 } \end{matrix} $$

(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) = 3x^4-11x^3-6x^2+8x } $, 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.

$$ \lim_{x \to -\infty} \left( 3x^4-11x^3-6x^2+8x \right) = \lim_{x \to -\infty} 3x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 3x^4-11x^3-6x^2+8x \right) = \lim_{x \to \infty} 3x^4 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 12x^3-33x^2-12x+8 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3588 & x_2 = -0.6175 & x_3 = 3.0087 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3588 } \Rightarrow p\left(0.3588\right) = \color{orangered}{ 1.6396 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.6175 } \Rightarrow p\left(-0.6175\right) = \color{orangered}{ -4.2016 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.0087 } \Rightarrow p\left(3.0087\right) = \color{orangered}{ -84.0044 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.3588, 1.6396 \right) & \left( -0.6175, -4.2016 \right) & \left( 3.0087, -84.0044 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 36x^2-66x-12 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 2 & x_2 = -\dfrac{ 1 }{ 6 } \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2 } \Rightarrow p\left(2\right) = \color{orangered}{ -48 }\\[1 em] \text{for } ~ x & = \color{blue}{ -\frac{ 1 }{ 6 } } \Rightarrow p\left(-\frac{ 1 }{ 6 }\right) = \color{orangered}{ -\frac{ 625 }{ 432 } }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 2, -48 \right) & \left( -\dfrac{ 1 }{ 6 }, -\dfrac{ 625 }{ 432 } \right)\end{matrix} $$

Graphing Polynomials