Question

g(t) = t^2 - t - 421) What are the zeros of the function?Write the smaller t first, and the larger t second.smaller t = -6larger t = 72) What is the vertex of the parabola?

Answer 1

$\begin{gathered} 1)\text{ smaller t = -6} \\ \text{larger t = 7} \\ 2)\text{ Vertex = (}\frac{1}{2},\frac{-169}{4}) \end{gathered}$

1) To get the zeros of the function, we have to equate the function to zero and solve for the roots of the quadratic equation

We have this as follows;

$\begin{gathered} t^2-t-42\text{ = 0} \\ t^2-7t+6t-42\text{ = 0} \\ t(t-7)+6(t-7)\text{ = 0} \\ (t+6)(t-7)\text{ = 0} \\ t\text{ + 6 = 0} \\ t\text{ -7 = 0} \\ \\ t\text{ = -6 ot t = 7} \end{gathered}$

smaller t is -6

Larger t is 7

2) We want to get the vertex of the parabola

To get this, we need to write thw function in the vertex form

We have the vertex form as;

$g(t)=a(t-h)^2\text{ + k}$

The vertex of the parabola is the point (h,k)

We can write the vertex form as follows;

$\begin{gathered} g(t)\text{ = (t-}\frac{1}{2})^2\text{ - }\frac{169}{4} \\ \\ h\text{ = }\frac{1}{2} \\ \\ k\text{ = }\frac{-169}{4} \end{gathered}$

The vertex of the parabola is thus;

$(\frac{1}{2},\text{ }\frac{-169}{4})$