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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.8471 & x_2 = 1.0309 \end{matrix} $$

(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^8-5x^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.

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

The graph starts in the upper-left corner.

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

The graph ends in the upper-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = 16x^7-35x^6+6x^5+25x^4+12x^3+6x^2+6x-5 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = 0.4128 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4128 } \Rightarrow p\left(0.4128\right) = \color{orangered}{ -8.2686 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 0.4128, -8.2686 \right)\end{matrix} $$

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^6-210x^5+30x^4+100x^3+36x^2+12x+6 $.

Since above equation has no solutions we conclude that
polynomial has no inflection points.

Graphing Polynomials