Sketch the graph of the polynomial function $$ p(x) = x^5+8x^4+17x^3-2x^2-24x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^5+8x^4+17x^3-2x^2-24x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 1 & x_3 = -2 & x_4 = -3 & x_5 = -4 \end{matrix} $$

(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) = x^5+8x^4+17x^3-2x^2-24x } $, so:

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

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^5+8x^4+17x^3-2x^2-24x \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5+8x^4+17x^3-2x^2-24x \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4+32x^3+51x^2-4x-24 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.605 & x_2 = -0.8738 & x_3 = -2.5022 & x_4 = -3.629 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.605 } \Rightarrow p\left(0.605\right) = \color{orangered}{ -10.3346 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8738 } \Rightarrow p\left(-0.8738\right) = \color{orangered}{ 12.2566 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.5022 } \Rightarrow p\left(-2.5022\right) = \color{orangered}{ -3.2813 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.629 } \Rightarrow p\left(-3.629\right) = \color{orangered}{ 6.3859 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.605, -10.3346 \right) & \left( -0.8738, 12.2566 \right) & \left( -2.5022, -3.2813 \right) & \left( -3.629, 6.3859 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 20x^3+96x^2+102x-4 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.0379 & x_2 = -1.6652 & x_3 = -3.1726 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.0379 } \Rightarrow p\left(0.0379\right) = \color{orangered}{ -0.9106 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.6652 } \Rightarrow p\left(-1.6652\right) = \color{orangered}{ 4.6304 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.1726 } \Rightarrow p\left(-3.1726\right) = \color{orangered}{ 2.2172 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.0379, -0.9106 \right) & \left( -1.6652, 4.6304 \right) & \left( -3.1726, 2.2172 \right)\end{matrix} $$

Graphing Polynomials