Sketch the graph of the polynomial function $$ p(x) = x^5+4x^4-8x^3-32x^2+16x+64 $$

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+4x^4-8x^3-32x^2+16x+64 = 0 } $

The solutions of this equation are:

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

(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+4x^4-8x^3-32x^2+16x+64 } $, so:

$$ \text{Y inercept} = p(0) = 64 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^5+4x^4-8x^3-32x^2+16x+64 \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5+4x^4-8x^3-32x^2+16x+64 \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4+16x^3-24x^2-64x+16 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 2 & x_2 = -2 & x_3 = 0.233 & x_4 = -3.433 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2 } \Rightarrow p\left(2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2 } \Rightarrow p\left(-2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.233 } \Rightarrow p\left(0.233\right) = \color{orangered}{ 65.902 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.433 } \Rightarrow p\left(-3.433\right) = \color{orangered}{ 34.368 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 2, 0 \right) & \left( -2, 0 \right) & \left( 0.233, 65.902 \right) & \left( -3.433, 34.368 \right)\end{matrix} $$

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+48x^2-48x-64 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.3075 & x_2 = -0.8593 & x_3 = -2.8482 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.3075 } \Rightarrow p\left(1.3075\right) = \color{orangered}{ 27.845 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8593 } \Rightarrow p\left(-0.8593\right) = \color{orangered}{ 33.4105 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.8482 } \Rightarrow p\left(-2.8482\right) = \color{orangered}{ 19.4765 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.3075, 27.845 \right) & \left( -0.8593, 33.4105 \right) & \left( -2.8482, 19.4765 \right)\end{matrix} $$

Graphing Polynomials