STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 3 & x_2 = 4 & x_3 = -2 & x_4 = -4 & x_5 = 3 & x_6 = -2 & x_7 = -4 & x_8 = -2 \end{matrix} $$

(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) = -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 } $, so:

$$ \text{Y inercept} = p(0) = 69120 $$

STEP 3:Find the end behavior

The end behavior of a polynomial is the same as the end behavior of a leading term.

$$ \lim_{x \to -\infty} \left( -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 \right) = \lim_{x \to -\infty} -15x^8 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -15x^8-60x^7+465x^6+2010x^5-3900x^4-21480x^3+480x^2+74880x+69120 \right) = \lim_{x \to \infty} -15x^8 = \color{blue}{ -\infty } $$

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) $:

$$ p^{\prime} (x) = -120x^7-420x^6+2790x^5+10050x^4-15600x^3-64440x^2+960x+74880 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 3 & x_2 = -2 & x_3 = -4 & x_4 = -2 & x_5 = 1.0644 & x_6 = -3.2838 & x_7 = 3.7194 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 3 } \Rightarrow p\left(3\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2 } \Rightarrow p\left(-2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -4 } \Rightarrow p\left(-4\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2 } \Rightarrow p\left(-2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.0644 } \Rightarrow p\left(1.0644\right) = \color{orangered}{ 121761.9869 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.2838 } \Rightarrow p\left(-3.2838\right) = \color{orangered}{ -4682.2514 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.7194 } \Rightarrow p\left(3.7194\right) = \color{orangered}{ 24285.0106 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 3, 0 \right) & \left( -2, 0 \right) & \left( -4, 0 \right) & \left( -2, 0 \right) & \left( 1.0644, 121761.9869 \right) & \left( -3.2838, -4682.2514 \right) & \left( 3.7194, 24285.0106 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -840x^6-2520x^5+13950x^4+40200x^3-46800x^2-128880x+960 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -2 & x_2 = 0.0074 & x_3 = -2.8221 & x_4 = 2.1271 & x_5 = 3.4269 & x_6 = -3.7393 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -2 } \Rightarrow p\left(-2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.0074 } \Rightarrow p\left(0.0074\right) = \color{orangered}{ 69676.3761 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.8221 } \Rightarrow p\left(-2.8221\right) = \color{orangered}{ -2673.7921 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.1271 } \Rightarrow p\left(2.1271\right) = \color{orangered}{ 56492.0641 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.4269 } \Rightarrow p\left(3.4269\right) = \color{orangered}{ 13809.0209 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.7393 } \Rightarrow p\left(-3.7393\right) = \color{orangered}{ -1885.5067 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -2, 0 \right) & \left( 0.0074, 69676.3761 \right) & \left( -2.8221, -2673.7921 \right) & \left( 2.1271, 56492.0641 \right) & \left( 3.4269, 13809.0209 \right) & \left( -3.7393, -1885.5067 \right)\end{matrix} $$

Graphing Polynomials