Sketch the graph of the polynomial function $$ p(x) = 10x^8-290x^7+3600x^6-24900x^5+104490x^4-270810x^3+419900x^2-352000x+120000 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 10x^8-290x^7+3600x^6-24900x^5+104490x^4-270810x^3+419900x^2-352000x+120000 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1 & x_2 = 2 & x_3 = 3 & x_4 = 4 & x_5 = 5 & x_6 = 4 & x_7 = 5 & x_8 = 5 \end{matrix} $$

(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^8-290x^7+3600x^6-24900x^5+104490x^4-270810x^3+419900x^2-352000x+120000 } $, so:

$$ \text{Y inercept} = p(0) = 120000 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 10x^8-290x^7+3600x^6-24900x^5+104490x^4-270810x^3+419900x^2-352000x+120000 \right) = \lim_{x \to -\infty} 10x^8 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 10x^8-290x^7+3600x^6-24900x^5+104490x^4-270810x^3+419900x^2-352000x+120000 \right) = \lim_{x \to \infty} 10x^8 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 80x^7-2030x^6+21600x^5-124500x^4+417960x^3-812430x^2+839800x-352000 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 4 & x_2 = 5 & x_3 = 5 & x_4 = 1.2839 & x_5 = 2.3276 & x_6 = 3.2951 & x_7 = 4.4685 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 4 } \Rightarrow p\left(4\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 5 } \Rightarrow p\left(5\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 5 } \Rightarrow p\left(5\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.2839 } \Rightarrow p\left(1.2839\right) = \color{orangered}{ -1320.7948 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.3276 } \Rightarrow p\left(2.3276\right) = \color{orangered}{ 156.1074 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.2951 } \Rightarrow p\left(3.2951\right) = \color{orangered}{ -21.5986 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.4685 } \Rightarrow p\left(4.4685\right) = \color{orangered}{ -4.1436 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 4, 0 \right) & \left( 5, 0 \right) & \left( 5, 0 \right) & \left( 1.2839, -1320.7948 \right) & \left( 2.3276, 156.1074 \right) & \left( 3.2951, -21.5986 \right) & \left( 4.4685, -4.1436 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 560x^6-12180x^5+108000x^4-498000x^3+1253880x^2-1624860x+839800 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 5 & x_2 = 1.5882 & x_3 = 2.6605 & x_4 = 3.572 & x_5 = 4.2178 & x_6 = 4.7116 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 5 } \Rightarrow p\left(5\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.5882 } \Rightarrow p\left(1.5882\right) = \color{orangered}{ -790.0577 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.6605 } \Rightarrow p\left(2.6605\right) = \color{orangered}{ 85.5473 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.572 } \Rightarrow p\left(3.572\right) = \color{orangered}{ -12.3385 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.2178 } \Rightarrow p\left(4.2178\right) = \color{orangered}{ -1.9724 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.7116 } \Rightarrow p\left(4.7116\right) = \color{orangered}{ -2.0923 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 5, 0 \right) & \left( 1.5882, -790.0577 \right) & \left( 2.6605, 85.5473 \right) & \left( 3.572, -12.3385 \right) & \left( 4.2178, -1.9724 \right) & \left( 4.7116, -2.0923 \right)\end{matrix} $$

Graphing Polynomials