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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 6x^7-2x^6+4x^5+2x^4+6x^3+9x^2+x+3 = 0 } $

The solution of this equation is:

$$ \begin{matrix}x = -0.9009 \end{matrix} $$

(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) = 6x^7-2x^6+4x^5+2x^4+6x^3+9x^2+x+3 } $, so:

$$ \text{Y inercept} = p(0) = 3 $$

STEP 3:Find the end behavior

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

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

The graph starts in the lower-left corner.

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

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) $:

$$ p^{\prime} (x) = 42x^6-12x^5+20x^4+8x^3+18x^2+18x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.059 & x_2 = -0.5885 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.059 } \Rightarrow p\left(-0.059\right) = \color{orangered}{ 2.9711 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.5885 } \Rightarrow p\left(-0.5885\right) = \color{orangered}{ 4.0334 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.059, 2.9711 \right) & \left( -0.5885, 4.0334 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 252x^5-60x^4+80x^3+24x^2+36x+18 $.

The zero of second derivative is

$$ \begin{matrix}x = -0.3816 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.3816 } \Rightarrow p\left(-0.3816\right) = \color{orangered}{ 3.5924 }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( -0.3816, 3.5924 \right)\end{matrix} $$

Graphing Polynomials