Sketch the graph of the polynomial function $$ p(x) = \frac{1}{3}x^2-\frac{2}{3} $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ \frac{1}{3}x^2-\frac{2}{3} = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = \sqrt{ 2 } & x_2 = -\sqrt{ 2 } \end{matrix} $$

(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) = \frac{1}{3}x^2-\frac{2}{3} } $, so:

$$ \text{Y inercept} = p(0) = -\frac{ 2 }{ 3 } $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( \frac{1}{3}x^2-\frac{2}{3} \right) = \lim_{x \to -\infty} \frac{1}{3}x^2 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( \frac{1}{3}x^2-\frac{2}{3} \right) = \lim_{x \to \infty} \frac{1}{3}x^2 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = \frac{2}{3}x $$

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

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

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ -\frac{ 2 }{ 3 } }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 0, -\dfrac{ 2 }{ 3 } \right)\end{matrix} $$

Graphing Polynomials