Sketch the graph of the polynomial function $$ p(x) = 6x^7-20x^6+4x^5+2x^4+7x^3+9x^2+x+3 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 6x^7-20x^6+4x^5+2x^4+7x^3+9x^2+x+3 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.7414 & x_2 = 1.2657 & x_3 = 3.0132 \end{matrix} $$

(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^7-20x^6+4x^5+2x^4+7x^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.

$$ \lim_{x \to -\infty} \left( 6x^7-20x^6+4x^5+2x^4+7x^3+9x^2+x+3 \right) = \lim_{x \to -\infty} 6x^7 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 6x^7-20x^6+4x^5+2x^4+7x^3+9x^2+x+3 \right) = \lim_{x \to \infty} 6x^7 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 42x^6-120x^5+20x^4+8x^3+21x^2+18x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.0596 & x_2 = -0.4558 & x_3 = 0.9374 & x_4 = 2.6091 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.0596 } \Rightarrow p\left(-0.0596\right) = \color{orangered}{ 2.9709 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.4558 } \Rightarrow p\left(-0.4558\right) = \color{orangered}{ 3.5549 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.9374 } \Rightarrow p\left(0.9374\right) = \color{orangered}{ 12.2975 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.6091 } \Rightarrow p\left(2.6091\right) = \color{orangered}{ -603.2841 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.0596, 2.9709 \right) & \left( -0.4558, 3.5549 \right) & \left( 0.9374, 12.2975 \right) & \left( 2.6091, -603.2841 \right)\end{matrix} $$

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^5-600x^4+80x^3+24x^2+42x+18 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = -0.2994 & x_2 = 0.6482 & x_3 = 2.1981 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.2994 } \Rightarrow p\left(-0.2994\right) = \color{orangered}{ 3.3103 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.6482 } \Rightarrow p\left(0.6482\right) = \color{orangered}{ 8.9526 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.1981 } \Rightarrow p\left(2.1981\right) = \color{orangered}{ -393.2815 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( -0.2994, 3.3103 \right) & \left( 0.6482, 8.9526 \right) & \left( 2.1981, -393.2815 \right)\end{matrix} $$

Graphing Polynomials