Sketch the graph of the polynomial function $$ p(x) = 32x^3-x^2-7x+6 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 32x^3-x^2-7x+6 = 0 } $

The solution of this equation is:

$$ \begin{matrix}x = -0.686 \end{matrix} $$

(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) = 32x^3-x^2-7x+6 } $, so:

$$ \text{Y inercept} = p(0) = 6 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 32x^3-x^2-7x+6 \right) = \lim_{x \to -\infty} 32x^3 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( 32x^3-x^2-7x+6 \right) = \lim_{x \to \infty} 32x^3 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 96x^2-2x-7 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.2806 & x_2 = -0.2598 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.2806 } \Rightarrow p\left(0.2806\right) = \color{orangered}{ 4.6641 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.2598 } \Rightarrow p\left(-0.2598\right) = \color{orangered}{ 7.19 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.2806, 4.6641 \right) & \left( -0.2598, 7.19 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 192x-2 $.

The zero of second derivative is

$$ \begin{matrix}x = \dfrac{ 1 }{ 96 } \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ \frac{ 1 }{ 96 } } \Rightarrow p\left(\frac{ 1 }{ 96 }\right) = \color{orangered}{ \frac{ 81935 }{ 13824 } }\end{aligned} $$

So the inflection point is:

$$ \begin{matrix} \left( \dfrac{ 1 }{ 96 }, \dfrac{ 81935 }{ 13824 } \right)\end{matrix} $$

Graphing Polynomials