Question

Find the vertex, roots, and equation of the function below.

Answer 1

Vertex: (-4,3)

roots: (-5,0), (-3,0)

Equation: y = -3 ( x + 5) ( x+3)

Explanation:

As can be seen from the graph given, the vertex of the graph is at (4, 3).

The roots of the function at (-5, 0) and (-3, 0).

The equation of the function can be constructed using the roots of the function.

We know that the equation of a parabola can be written as

`y=k(x+a)(x+b)`

where roots of the function are at (-a, 0 ) and (-b, 0). From this, we deduce that for our function the equation must be

`y=k(x+5)(x+3)`

where k is a constant hitherto unknown.

We find the value of k using the point (-4, 3) from which we know that when x = -4, y= 3.

Putting in x = -4 and y = 3 in the above equation gives

`3=k(-4+5)(-4+3)`
`3=k(1)(-1)`
`3=-k`
`\begin{gathered} \boxed{\therefore k=-3} \\ \end{gathered}`

Hence, our equation is

`y=-3(x+5)(x+3)`

which is our answer!