Sketch the graph of the polynomial function $$ p(x) = \frac{6}{4}x^4+\frac{10}{3}x^3-2x^2+6x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ \frac{6}{4}x^4+\frac{10}{3}x^3-2x^2+6x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = -3.0777 \end{matrix} $$

(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) = \frac{6}{4}x^4+\frac{10}{3}x^3-2x^2+6x } $, 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.

$$ \lim_{x \to -\infty} \left( \frac{6}{4}x^4+\frac{10}{3}x^3-2x^2+6x \right) = \lim_{x \to -\infty} \frac{6}{4}x^4 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( \frac{6}{4}x^4+\frac{10}{3}x^3-2x^2+6x \right) = \lim_{x \to \infty} \frac{6}{4}x^4 = \color{blue}{ \infty } $$

The graph ends in the upper-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = 6x^3+10x^2-4x+6 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -2.1822 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -2.1822 } \Rightarrow p\left(-2.1822\right) = \color{orangered}{ -23.241 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( -2.1822, -23.241 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 18x^2+20x-4 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.173 & x_2 = -1.2842 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.173 } \Rightarrow p\left(0.173\right) = \color{orangered}{ 0.997 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.2842 } \Rightarrow p\left(-1.2842\right) = \color{orangered}{ -13.9828 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.173, 0.997 \right) & \left( -1.2842, -13.9828 \right)\end{matrix} $$

Graphing Polynomials