Sketch the graph of the polynomial function $$ p(x) = x^5-5x^4+6x^3-3x^2+5x-6 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^5-5x^4+6x^3-3x^2+5x-6 = 0 } $

The solution of this equation is:

$$ \begin{matrix}x = 3.4217 \end{matrix} $$

(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^5-5x^4+6x^3-3x^2+5x-6 } $, so:

$$ \text{Y inercept} = p(0) = -6 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^5-5x^4+6x^3-3x^2+5x-6 \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5-5x^4+6x^3-3x^2+5x-6 \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4-20x^3+18x^2-6x+5 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 1.1669 & x_2 = 2.8359 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.1669 } \Rightarrow p\left(1.1669\right) = \color{orangered}{ -1.8239 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.8359 } \Rightarrow p\left(2.8359\right) = \color{orangered}{ -19.076 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 1.1669, -1.8239 \right) & \left( 2.8359, -19.076 \right)\end{matrix} $$

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^3-60x^2+36x-6 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.3123 & x_2 = 0.4245 & x_3 = 2.2633 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3123 } \Rightarrow p\left(0.3123\right) = \color{orangered}{ -4.593 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.4245 } \Rightarrow p\left(0.4245\right) = \color{orangered}{ -4.1078 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.2633 } \Rightarrow p\left(2.2633\right) = \color{orangered}{ -12.2991 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.3123, -4.593 \right) & \left( 0.4245, -4.1078 \right) & \left( 2.2633, -12.2991 \right)\end{matrix} $$

Graphing Polynomials