Sketch the graph of the polynomial function $$ p(x) = 396-85x+150x^2-51x^3+8x^4-\frac{1}{2}x^5 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 396-85x+150x^2-51x^3+8x^4-\frac{1}{2}x^5 = 0 } $

The solution of this equation is:

$$ \begin{matrix}x = 7.6402 \end{matrix} $$

(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) = 396-85x+150x^2-51x^3+8x^4-\frac{1}{2}x^5 } $, so:

$$ \text{Y inercept} = p(0) = 396 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 396-85x+150x^2-51x^3+8x^4-\frac{1}{2}x^5 \right) = \lim_{x \to -\infty} 396 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 396-85x+150x^2-51x^3+8x^4-\frac{1}{2}x^5 \right) = \lim_{x \to \infty} 396 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = -\frac{5}{2}x^4+32x^3-153x^2+300x-85 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3374 & x_2 = 5.2883 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3374 } \Rightarrow p\left(0.3374\right) = \color{orangered}{ 382.5394 }\\[1 em] \text{for } ~ x & = \color{blue}{ 5.2883 } \Rightarrow p\left(5.2883\right) = \color{orangered}{ 787.6881 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.3374, 382.5394 \right) & \left( 5.2883, 787.6881 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -10x^3+96x^2-306x+300 $.

The zero of second derivative is

$$ \begin{matrix}x = 1.8342 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.8342 } \Rightarrow p\left(1.8342\right) = \color{orangered}{ 510.1917 }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( 1.8342, 510.1917 \right)\end{matrix} $$

Graphing Polynomials