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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1.2401 & x_2 = -1.7032 \end{matrix} $$

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

$$ \text{Y inercept} = p(0) = 2 $$

STEP 3:Find the end behavior

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

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

The graph starts in the lower-left corner.

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

The graph ends in the lower-right corner.

STEP 4:Find the turning points

To determine the turning points, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = -6x^5-5x^4+4x^3+3x^2+2x-1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3186 & x_2 = 0.8542 & x_3 = -1.2734 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3186 } \Rightarrow p\left(0.3186\right) = \color{orangered}{ 1.8212 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.8542 } \Rightarrow p\left(0.8542\right) = \color{orangered}{ 2.1879 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.2734 } \Rightarrow p\left(-1.2734\right) = \color{orangered}{ 4.5441 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.3186, 1.8212 \right) & \left( 0.8542, 2.1879 \right) & \left( -1.2734, 4.5441 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -30x^4-20x^3+12x^2+6x+2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.6533 & x_2 = -0.945 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.6533 } \Rightarrow p\left(0.6533\right) = \color{orangered}{ 2.0377 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.945 } \Rightarrow p\left(-0.945\right) = \color{orangered}{ 3.833 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.6533, 2.0377 \right) & \left( -0.945, 3.833 \right)\end{matrix} $$

Graphing Polynomials