Sketch the graph of the polynomial function $$ p(x) = -2x^5+19x^4-50x^3-21x^2+234x-216 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -2x^5+19x^4-50x^3-21x^2+234x-216 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 3 & x_2 = 4 & x_3 = -2 & x_4 = \dfrac{ 3 }{ 2 } & x_5 = 3 \end{matrix} $$

(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^5+19x^4-50x^3-21x^2+234x-216 } $, so:

$$ \text{Y inercept} = p(0) = -216 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( -2x^5+19x^4-50x^3-21x^2+234x-216 \right) = \lim_{x \to -\infty} -2x^5 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( -2x^5+19x^4-50x^3-21x^2+234x-216 \right) = \lim_{x \to \infty} -2x^5 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = -10x^4+76x^3-150x^2-42x+234 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 3 & x_2 = -1.0717 & x_3 = 1.9619 & x_4 = 3.7098 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 3 } \Rightarrow p\left(3\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.0717 } \Rightarrow p\left(-1.0717\right) = \color{orangered}{ -401.4614 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.9619 } \Rightarrow p\left(1.9619\right) = \color{orangered}{ 8.0387 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.7098 } \Rightarrow p\left(3.7098\right) = \color{orangered}{ 3.6895 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 3, 0 \right) & \left( -1.0717, -401.4614 \right) & \left( 1.9619, 8.0387 \right) & \left( 3.7098, 3.6895 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -40x^3+228x^2-300x-42 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -0.1274 & x_2 = 2.4164 & x_3 = 3.411 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.1274 } \Rightarrow p\left(-0.1274\right) = \color{orangered}{ -246.0416 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.4164 } \Rightarrow p\left(2.4164\right) = \color{orangered}{ 4.3653 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.411 } \Rightarrow p\left(3.411\right) = \color{orangered}{ 2.0572 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -0.1274, -246.0416 \right) & \left( 2.4164, 4.3653 \right) & \left( 3.411, 2.0572 \right)\end{matrix} $$

Graphing Polynomials