Sketch the graph of the polynomial function $$ p(x) = -\frac{1}{5}x^6-\frac{1}{4}x^5+\frac{1}{3}x^4+\frac{1}{2}x^3+x^2-x+2 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -\frac{1}{5}x^6-\frac{1}{4}x^5+\frac{1}{3}x^4+\frac{1}{2}x^3+x^2-x+2 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1.7037 & x_2 = -2.275 \end{matrix} $$

(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) = -\frac{1}{5}x^6-\frac{1}{4}x^5+\frac{1}{3}x^4+\frac{1}{2}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.

$$ \lim_{x \to -\infty} \left( -\frac{1}{5}x^6-\frac{1}{4}x^5+\frac{1}{3}x^4+\frac{1}{2}x^3+x^2-x+2 \right) = \lim_{x \to -\infty} -\frac{1}{5}x^6 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -\frac{1}{5}x^6-\frac{1}{4}x^5+\frac{1}{3}x^4+\frac{1}{2}x^3+x^2-x+2 \right) = \lim_{x \to \infty} -\frac{1}{5}x^6 = \color{blue}{ -\infty } $$

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) $:

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

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3757 & x_2 = 1.2274 & x_3 = -1.701 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3757 } \Rightarrow p\left(0.3757\right) = \color{orangered}{ 1.7962 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.2274 } \Rightarrow p\left(1.2274\right) = \color{orangered}{ 2.5799 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.701 } \Rightarrow p\left(-1.701\right) = \color{orangered}{ 5.6397 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.3757, 1.7962 \right) & \left( 1.2274, 2.5799 \right) & \left( -1.701, 5.6397 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -6x^4-5x^3+4x^2+3x+2 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.9178 & x_2 = -1.2255 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.9178 } \Rightarrow p\left(0.9178\right) = \color{orangered}{ 2.2653 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.2255 } \Rightarrow p\left(-1.2255\right) = \color{orangered}{ 4.5725 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.9178, 2.2653 \right) & \left( -1.2255, 4.5725 \right)\end{matrix} $$

Graphing Polynomials