STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ \frac{105}{1000000}x^3-\frac{8}{10000}x^2-\frac{75}{100}x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 88.4108 & x_3 = -80.7917 \end{matrix} $$

(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{105}{1000000}x^3-\frac{8}{10000}x^2-\frac{75}{100}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.

$$ \lim_{x \to -\infty} \left( \frac{105}{1000000}x^3-\frac{8}{10000}x^2-\frac{75}{100}x \right) = \lim_{x \to -\infty} \frac{105}{1000000}x^3 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( \frac{105}{1000000}x^3-\frac{8}{10000}x^2-\frac{75}{100}x \right) = \lim_{x \to \infty} \frac{105}{1000000}x^3 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = \frac{63}{200000}x^2-\frac{1}{625}x-\frac{3}{4} $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 51.4007 & x_2 = -46.3214 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 51.4007 } \Rightarrow p\left(51.4007\right) = \color{orangered}{ -26.4049 }\\[1 em] \text{for } ~ x & = \color{blue}{ -46.3214 } \Rightarrow p\left(-46.3214\right) = \color{orangered}{ 22.5885 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 51.4007, -26.4049 \right) & \left( -46.3214, 22.5885 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = \frac{63}{100000}x-\frac{1}{625} $.

The zero of second derivative is

$$ \begin{matrix}x = \dfrac{ 160 }{ 63 } \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ \frac{ 160 }{ 63 } } \Rightarrow p\left(\frac{ 160 }{ 63 }\right) = \color{orangered}{ -\frac{ 568024 }{ 297675 } }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( \dfrac{ 160 }{ 63 }, -\dfrac{ 568024 }{ 297675 } \right)\end{matrix} $$

Graphing Polynomials