Sketch the graph of the polynomial function $$ p(x) = 2x^6-59x^5+664x^4-3752x^3+11063x^2-15374x+17469 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^6-59x^5+664x^4-3752x^3+11063x^2-15374x+17469 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 7.3412 & x_2 = 11.9748 \end{matrix} $$

(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) = 2x^6-59x^5+664x^4-3752x^3+11063x^2-15374x+17469 } $, so:

$$ \text{Y inercept} = p(0) = 17469 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 2x^6-59x^5+664x^4-3752x^3+11063x^2-15374x+17469 \right) = \lim_{x \to -\infty} 2x^6 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 2x^6-59x^5+664x^4-3752x^3+11063x^2-15374x+17469 \right) = \lim_{x \to \infty} 2x^6 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 12x^5-295x^4+2656x^3-11256x^2+22126x-15374 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 1.481 & x_2 = 3.644 & x_3 = 10.6211 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.481 } \Rightarrow p\left(1.481\right) = \color{orangered}{ 9572.5142 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.644 } \Rightarrow p\left(3.644\right) = \color{orangered}{ 10651.3931 }\\[1 em] \text{for } ~ x & = \color{blue}{ 10.6211 } \Rightarrow p\left(10.6211\right) = \color{orangered}{ -46801.4293 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 1.481, 9572.5142 \right) & \left( 3.644, 10651.3931 \right) & \left( 10.6211, -46801.4293 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 60x^4-1180x^3+7968x^2-22512x+22126 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 2.2028 & x_2 = 9.1631 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2.2028 } \Rightarrow p\left(2.2028\right) = \color{orangered}{ 9982.7917 }\\[1 em] \text{for } ~ x & = \color{blue}{ 9.1631 } \Rightarrow p\left(9.1631\right) = \color{orangered}{ -27565.2845 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 2.2028, 9982.7917 \right) & \left( 9.1631, -27565.2845 \right)\end{matrix} $$

Graphing Polynomials