Sketch the graph of the polynomial function $$ p(x) = x^5-5x^4+4x^3-7x^2+5x-6 $$

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+4x^3-7x^2+5x-6 = 0 } $

The solution of this equation is:

$$ \begin{matrix}x = 4.4105 \end{matrix} $$

(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+4x^3-7x^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.

$$ \lim_{x \to -\infty} \left( x^5-5x^4+4x^3-7x^2+5x-6 \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5-5x^4+4x^3-7x^2+5x-6 \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4-20x^3+12x^2-14x+5 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.4131 & x_2 = 3.5214 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4131 } \Rightarrow p\left(0.4131\right) = \color{orangered}{ -4.9807 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.5214 } \Rightarrow p\left(3.5214\right) = \color{orangered}{ -127.8887 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.4131, -4.9807 \right) & \left( 3.5214, -127.8887 \right)\end{matrix} $$

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+24x-14 $.

The zero of second derivative is

$$ \begin{matrix}x = 2.6465 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2.6465 } \Rightarrow p\left(2.6465\right) = \color{orangered}{ -83.1048 }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( 2.6465, -83.1048 \right)\end{matrix} $$

Graphing Polynomials