Sketch the graph of the polynomial function $$ p(x) = -\frac{5}{12}x^4-\frac{1}{3}x^3-\frac{7}{12}x^2+\frac{5}{6}x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -\frac{5}{12}x^4-\frac{1}{3}x^3-\frac{7}{12}x^2+\frac{5}{6}x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 0.768 \end{matrix} $$

(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{5}{12}x^4-\frac{1}{3}x^3-\frac{7}{12}x^2+\frac{5}{6}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.

$$ \lim_{x \to -\infty} \left( -\frac{5}{12}x^4-\frac{1}{3}x^3-\frac{7}{12}x^2+\frac{5}{6}x \right) = \lim_{x \to -\infty} -\frac{5}{12}x^4 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -\frac{5}{12}x^4-\frac{1}{3}x^3-\frac{7}{12}x^2+\frac{5}{6}x \right) = \lim_{x \to \infty} -\frac{5}{12}x^4 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = -\frac{5}{3}x^3-x^2-\frac{7}{6}x+\frac{5}{6} $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = 0.4348 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4348 } \Rightarrow p\left(0.4348\right) = \color{orangered}{ 0.2098 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 0.4348, 0.2098 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -5x^2-2x-\frac{7}{6} $.

Since above equation has no solutions we conclude that
polynomial has no inflection points.

Graphing Polynomials