Sketch the graph of the polynomial function $$ p(x) = 3x^9-7x^8-4x^6+5x^5-3x+8 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 3x^9-7x^8-4x^6+5x^5-3x+8 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1.0543 & x_2 = -0.9204 & x_3 = 2.4423 \end{matrix} $$

(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) = 3x^9-7x^8-4x^6+5x^5-3x+8 } $, so:

$$ \text{Y inercept} = p(0) = 8 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 3x^9-7x^8-4x^6+5x^5-3x+8 \right) = \lim_{x \to -\infty} 3x^9 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 3x^9-7x^8-4x^6+5x^5-3x+8 \right) = \lim_{x \to \infty} 3x^9 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 27x^8-56x^7-24x^5+25x^4-3 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.5051 & x_2 = 2.1726 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.5051 } \Rightarrow p\left(-0.5051\right) = \color{orangered}{ 9.2484 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.1726 } \Rightarrow p\left(2.1726\right) = \color{orangered}{ -416.5228 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.5051, 9.2484 \right) & \left( 2.1726, -416.5228 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 216x^7-392x^6-120x^4+100x^3 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0 & x_2 = 0.5145 & x_3 = 1.9012 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 8 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5145 } \Rightarrow p\left(0.5145\right) = \color{orangered}{ 6.5357 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.9012 } \Rightarrow p\left(1.9012\right) = \color{orangered}{ -283.6602 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0, 8 \right) & \left( 0.5145, 6.5357 \right) & \left( 1.9012, -283.6602 \right)\end{matrix} $$

Graphing Polynomials