Sketch the graph of the polynomial function $$ p(x) = x^8-18x^6-320x^4+11x^7-472x^5+6576x^3-27648x+7776x^2-41472 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^8-18x^6-320x^4+11x^7-472x^5+6576x^3-27648x+7776x^2-41472 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 3 & x_2 = 4 & x_3 = -2 & x_4 = -6 & x_5 = 4 & x_6 = -2 & x_7 = -6 & x_8 = -6 \end{matrix} $$

(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) = x^8-18x^6-320x^4+11x^7-472x^5+6576x^3-27648x+7776x^2-41472 } $, so:

$$ \text{Y inercept} = p(0) = -41472 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^8-18x^6-320x^4+11x^7-472x^5+6576x^3-27648x+7776x^2-41472 \right) = \lim_{x \to -\infty} x^8 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^8-18x^6-320x^4+11x^7-472x^5+6576x^3-27648x+7776x^2-41472 \right) = \lim_{x \to \infty} x^8 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 8x^7+77x^6-108x^5-2360x^4-1280x^3+19728x^2+15552x-27648 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 4 & x_2 = -2 & x_3 = -6 & x_4 = -6 & x_5 = 0.9042 & x_6 = 3.388 & x_7 = -3.9172 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 4 } \Rightarrow p\left(4\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2 } \Rightarrow p\left(-2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -6 } \Rightarrow p\left(-6\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -6 } \Rightarrow p\left(-6\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.9042 } \Rightarrow p\left(0.9042\right) = \color{orangered}{ -55755.6342 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.388 } \Rightarrow p\left(3.388\right) = \color{orangered}{ 3490.6716 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.9172 } \Rightarrow p\left(-3.9172\right) = \color{orangered}{ -14399.6227 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 4, 0 \right) & \left( -2, 0 \right) & \left( -6, 0 \right) & \left( -6, 0 \right) & \left( 0.9042, -55755.6342 \right) & \left( 3.388, 3490.6716 \right) & \left( -3.9172, -14399.6227 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 56x^6+462x^5-540x^4-9440x^3-3840x^2+39456x+15552 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -6 & x_2 = -0.3932 & x_3 = -2.9197 & x_4 = 2.2115 & x_5 = 3.7338 & x_6 = -4.8823 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -6 } \Rightarrow p\left(-6\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.3932 } \Rightarrow p\left(-0.3932\right) = \color{orangered}{ -29800.7251 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.9197 } \Rightarrow p\left(-2.9197\right) = \color{orangered}{ -7007.567 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.2115 } \Rightarrow p\left(2.2115\right) = \color{orangered}{ -24769.6967 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.7338 } \Rightarrow p\left(3.7338\right) = \color{orangered}{ 1576.8766 }\\[1 em] \text{for } ~ x & = \color{blue}{ -4.8823 } \Rightarrow p\left(-4.8823\right) = \color{orangered}{ -7213.5797 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -6, 0 \right) & \left( -0.3932, -29800.7251 \right) & \left( -2.9197, -7007.567 \right) & \left( 2.2115, -24769.6967 \right) & \left( 3.7338, 1576.8766 \right) & \left( -4.8823, -7213.5797 \right)\end{matrix} $$

Graphing Polynomials