Sketch the graph of the polynomial function $$ p(x) = x^6+2x^5+2x^4+7x^3+9x^2-x+4 $$

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+2x^5+2x^4+7x^3+9x^2-x+4 = 0 } $

Since above equation has no solutions we conclude that
polynomial has no x-intecepts.

(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+2x^5+2x^4+7x^3+9x^2-x+4 } $, so:

$$ \text{Y inercept} = p(0) = 4 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^6+2x^5+2x^4+7x^3+9x^2-x+4 \right) = \lim_{x \to -\infty} x^6 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( x^6+2x^5+2x^4+7x^3+9x^2-x+4 \right) = \lim_{x \to \infty} x^6 = \color{blue}{ \infty } $$

The graph ends in the upper-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = 6x^5+10x^4+8x^3+21x^2+18x-1 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = 0.0523 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.0523 } \Rightarrow p\left(0.0523\right) = \color{orangered}{ 3.9733 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 0.0523, 3.9733 \right)\end{matrix} $$

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+40x^3+24x^2+42x+18 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -0.4962 & x_2 = -1.2772 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.4962 } \Rightarrow p\left(-0.4962\right) = \color{orangered}{ 5.933 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.2772 } \Rightarrow p\left(-1.2772\right) = \color{orangered}{ 8.2398 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -0.4962, 5.933 \right) & \left( -1.2772, 8.2398 \right)\end{matrix} $$

Graphing Polynomials