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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 2x^8+3x^7+x^6+5x^5+3x^4+2x^3+3x^2-5x-7 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0.8974 & x_2 = -1.7488 \end{matrix} $$

(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) = 2x^8+3x^7+x^6+5x^5+3x^4+2x^3+3x^2-5x-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.

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

The graph starts in the upper-left corner.

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

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) $:

$$ p^{\prime} (x) = 16x^7+21x^6+6x^5+25x^4+12x^3+6x^2+6x-5 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.4023 & x_2 = -0.8488 & x_3 = -1.5258 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4023 } \Rightarrow p\left(0.4023\right) = \color{orangered}{ -8.2537 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8488 } \Rightarrow p\left(-0.8488\right) = \color{orangered}{ -2.5028 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.5258 } \Rightarrow p\left(-1.5258\right) = \color{orangered}{ -10.9685 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.4023, -8.2537 \right) & \left( -0.8488, -2.5028 \right) & \left( -1.5258, -10.9685 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 112x^6+126x^5+30x^4+100x^3+36x^2+12x+6 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -0.4145 & x_2 = -1.3176 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.4145 } \Rightarrow p\left(-0.4145\right) = \color{orangered}{ -4.5264 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.3176 } \Rightarrow p\left(-1.3176\right) = \color{orangered}{ -7.8745 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -0.4145, -4.5264 \right) & \left( -1.3176, -7.8745 \right)\end{matrix} $$

Graphing Polynomials