Sketch the graph of the polynomial function $$ p(x) = 8x^6+8x^5-9x^4-6x^3+2x^2+7 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 8x^6+8x^5-9x^4-6x^3+2x^2+7 = 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) = 8x^6+8x^5-9x^4-6x^3+2x^2+7 } $, so:

$$ \text{Y inercept} = p(0) = 7 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 8x^6+8x^5-9x^4-6x^3+2x^2+7 \right) = \lim_{x \to -\infty} 8x^6 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 8x^6+8x^5-9x^4-6x^3+2x^2+7 \right) = \lim_{x \to \infty} 8x^6 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 48x^5+40x^4-36x^3-18x^2+4x $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = 0.1753 & x_3 = -0.5784 & x_4 = 0.7167 & x_5 = -1.147 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 7 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.1753 } \Rightarrow p\left(0.1753\right) = \color{orangered}{ 7.0222 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.5784 } \Rightarrow p\left(-0.5784\right) = \color{orangered}{ 7.6045 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.7167 } \Rightarrow p\left(0.7167\right) = \color{orangered}{ 6.0409 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.147 } \Rightarrow p\left(-1.147\right) = \color{orangered}{ 5.4425 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0, 7 \right) & \left( 0.1753, 7.0222 \right) & \left( -0.5784, 7.6045 \right) & \left( 0.7167, 6.0409 \right) & \left( -1.147, 5.4425 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 240x^4+160x^3-108x^2-36x+4 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.0904 & x_2 = -0.3519 & x_3 = 0.5493 & x_4 = -0.9543 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.0904 } \Rightarrow p\left(0.0904\right) = \color{orangered}{ 7.0114 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.3519 } \Rightarrow p\left(-0.3519\right) = \color{orangered}{ 7.3432 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5493 } \Rightarrow p\left(0.5493\right) = \color{orangered}{ 6.4096 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.9543 } \Rightarrow p\left(-0.9543\right) = \color{orangered}{ 6.2821 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.0904, 7.0114 \right) & \left( -0.3519, 7.3432 \right) & \left( 0.5493, 6.4096 \right) & \left( -0.9543, 6.2821 \right)\end{matrix} $$

Graphing Polynomials