Sketch the graph of the polynomial function $$ p(x) = -2x^4+39x^3-285x^2+766x+85 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -2x^4+39x^3-285x^2+766x+85 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.1067 & x_2 = 7.2296 \end{matrix} $$

(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^4+39x^3-285x^2+766x+85 } $, so:

$$ \text{Y inercept} = p(0) = 85 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( -2x^4+39x^3-285x^2+766x+85 \right) = \lim_{x \to -\infty} -2x^4 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -2x^4+39x^3-285x^2+766x+85 \right) = \lim_{x \to \infty} -2x^4 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

To determine the turning point, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = -8x^3+117x^2-570x+766 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = 2.1603 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2.1603 } \Rightarrow p\left(2.1603\right) = \color{orangered}{ 759.3585 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 2.1603, 759.3585 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -24x^2+234x-570 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 5 & x_2 = \dfrac{ 19 }{ 4 } \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 5 } \Rightarrow p\left(5\right) = \color{orangered}{ 415 }\\[1 em] \text{for } ~ x & = \color{blue}{ \frac{ 19 }{ 4 } } \Rightarrow p\left(\frac{ 19 }{ 4 }\right) = \color{orangered}{ \frac{ 58209 }{ 128 } }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 5, 415 \right) & \left( \dfrac{ 19 }{ 4 }, \dfrac{ 58209 }{ 128 } \right)\end{matrix} $$

Graphing Polynomials