Sketch the graph of the polynomial function $$ p(x) = x^5-8x^4+3x^3+82x^2-78x $$

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+3x^3+82x^2-78x = 0 } $

The solutions of this equation are:

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

(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) = x^5-8x^4+3x^3+82x^2-78x } $, 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.

$$ \lim_{x \to -\infty} \left( x^5-8x^4+3x^3+82x^2-78x \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5-8x^4+3x^3+82x^2-78x \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4-32x^3+9x^2+164x-78 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.4831 & x_2 = -2.0771 & x_3 = 3.3409 & x_4 = 4.6531 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4831 } \Rightarrow p\left(0.4831\right) = \color{orangered}{ -18.6154 }\\[1 em] \text{for } ~ x & = \color{blue}{ -2.0771 } \Rightarrow p\left(-2.0771\right) = \color{orangered}{ 301.3353 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.3409 } \Rightarrow p\left(3.3409\right) = \color{orangered}{ 186.0934 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.6531 } \Rightarrow p\left(4.6531\right) = \color{orangered}{ 145.7441 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.4831, -18.6154 \right) & \left( -2.0771, 301.3353 \right) & \left( 3.3409, 186.0934 \right) & \left( 4.6531, 145.7441 \right)\end{matrix} $$

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+18x+164 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -1.1047 & x_2 = 1.815 & x_3 = 4.0897 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -1.1047 } \Rightarrow p\left(-1.1047\right) = \color{orangered}{ 168.6304 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.815 } \Rightarrow p\left(1.815\right) = \color{orangered}{ 79.3789 }\\[1 em] \text{for } ~ x & = \color{blue}{ 4.0897 } \Rightarrow p\left(4.0897\right) = \color{orangered}{ 163.8213 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -1.1047, 168.6304 \right) & \left( 1.815, 79.3789 \right) & \left( 4.0897, 163.8213 \right)\end{matrix} $$

Graphing Polynomials