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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = -1.6629 \end{matrix} $$

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

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 10x^3+2x^2-8x+3 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -1.1363 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -1.1363 } \Rightarrow p\left(-1.1363\right) = \color{orangered}{ -5.3839 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( -1.1363, -5.3839 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 30x^2+4x-8 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.454 & x_2 = -0.5873 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.454 } \Rightarrow p\left(0.454\right) = \color{orangered}{ 0.7061 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.5873 } \Rightarrow p\left(-0.5873\right) = \color{orangered}{ -2.9795 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.454, 0.7061 \right) & \left( -0.5873, -2.9795 \right)\end{matrix} $$

Graphing Polynomials