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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ x^4-x^3-7x^2+x+6 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1 & x_2 = 3 & x_3 = -1 & x_4 = -2 \end{matrix} $$

(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^4-x^3-7x^2+x+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.

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 4x^3-3x^2-14x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.0705 & x_2 = -1.5742 & x_3 = 2.2538 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.0705 } \Rightarrow p\left(0.0705\right) = \color{orangered}{ 6.0354 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.5742 } \Rightarrow p\left(-1.5742\right) = \color{orangered}{ -2.8789 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.2538 } \Rightarrow p\left(2.2538\right) = \color{orangered}{ -12.9495 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.0705, 6.0354 \right) & \left( -1.5742, -2.8789 \right) & \left( 2.2538, -12.9495 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 12x^2-6x-14 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.3587 & x_2 = -0.8587 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.3587 } \Rightarrow p\left(1.3587\right) = \color{orangered}{ -4.6638 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8587 } \Rightarrow p\left(-0.8587\right) = \color{orangered}{ 1.1568 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.3587, -4.6638 \right) & \left( -0.8587, 1.1568 \right)\end{matrix} $$

Graphing Polynomials