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

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.289 & x_2 = -1.7708 \end{matrix} $$

(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^3-7x^2+5x+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.

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 6x^5+5x^4+4x^3-3x^2-14x+5 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3508 & x_2 = -1.3231 & x_3 = 0.9114 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3508 } \Rightarrow p\left(0.3508\right) = \color{orangered}{ 2.8717 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.3231 } \Rightarrow p\left(-1.3231\right) = \color{orangered}{ -10.1788 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.9114 } \Rightarrow p\left(0.9114\right) = \color{orangered}{ 1.8774 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.3508, 2.8717 \right) & \left( -1.3231, -10.1788 \right) & \left( 0.9114, 1.8774 \right)\end{matrix} $$

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-14 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.6777 & x_2 = -0.7889 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.6777 } \Rightarrow p\left(0.6777\right) = \color{orangered}{ 2.313 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.7889 } \Rightarrow p\left(-0.7889\right) = \color{orangered}{ -5.4866 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.6777, 2.313 \right) & \left( -0.7889, -5.4866 \right)\end{matrix} $$

Graphing Polynomials