STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -10x^5-12x^4+18x^3 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 0.8697 & x_3 = -2.0697 \end{matrix} $$

(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) = -10x^5-12x^4+18x^3 } $, 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.

$$ \lim_{x \to -\infty} \left( -10x^5-12x^4+18x^3 \right) = \lim_{x \to -\infty} -10x^5 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( -10x^5-12x^4+18x^3 \right) = \lim_{x \to \infty} -10x^5 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = -50x^4-48x^3+54x^2 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = 0.6647 & x_3 = -1.6247 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.6647 } \Rightarrow p\left(0.6647\right) = \color{orangered}{ 1.6462 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.6247 } \Rightarrow p\left(-1.6247\right) = \color{orangered}{ -47.6035 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0, 0 \right) & \left( 0.6647, 1.6462 \right) & \left( -1.6247, -47.6035 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -200x^3-144x^2+108x $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0 & x_2 = 0.4583 & x_3 = -1.1783 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.4583 } \Rightarrow p\left(0.4583\right) = \color{orangered}{ 1.0011 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.1783 } \Rightarrow p\left(-1.1783\right) = \color{orangered}{ -29.8647 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0, 0 \right) & \left( 0.4583, 1.0011 \right) & \left( -1.1783, -29.8647 \right)\end{matrix} $$

Graphing Polynomials