Sketch the graph of the polynomial function $$ p(s) = 4s^2-4s+1 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ 4s^2-4s+1 = 0 } $

The solution of this equation is:

$$ \begin{matrix}s = \dfrac{ 1 }{ 2 } \end{matrix} $$

(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 $ s = 0 $ into $ \color{blue}{ p(s) = 4s^2-4s+1 } $, so:

$$ \text{Y inercept} = p(0) = 1 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( 4s^2-4s+1 \right) = \lim_{x \to -\infty} 4s^2 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( 4s^2-4s+1 \right) = \lim_{x \to \infty} 4s^2 = \color{blue}{ \infty } $$

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(s) $:

$$ p^{\prime} (x) = 8s-4 $$

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

$$ p^{\prime} (s) = 0 $$ $$ \begin{matrix}s = \dfrac{ 1 }{ 2 } \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ s & = \color{blue}{ \frac{ 1 }{ 2 } } \Rightarrow p\left(\frac{ 1 }{ 2 }\right) = \color{orangered}{ 0 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( \dfrac{ 1 }{ 2 }, 0 \right)\end{matrix} $$

Graphing Polynomials