STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -5t^2+24t+30 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}t_1 = 5.8293 & t_2 = -1.0293 \end{matrix} $$

(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 $ t = 0 $ into $ \color{blue}{ p(t) = -5t^2+24t+30 } $, so:

$$ \text{Y inercept} = p(0) = 30 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( -5t^2+24t+30 \right) = \lim_{x \to -\infty} -5t^2 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -5t^2+24t+30 \right) = \lim_{x \to \infty} -5t^2 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

To determine the turning point, we need to find the first derivative of $ p(t) $:

$$ p^{\prime} (x) = -10t+24 $$

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

$$ p^{\prime} (t) = 0 $$ $$ \begin{matrix}t = \dfrac{ 12 }{ 5 } \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ t & = \color{blue}{ \frac{ 12 }{ 5 } } \Rightarrow p\left(\frac{ 12 }{ 5 }\right) = \color{orangered}{ \frac{ 294 }{ 5 } }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( \dfrac{ 12 }{ 5 }, \dfrac{ 294 }{ 5 } \right)\end{matrix} $$

Graphing Polynomials