a firework is modelled by the relation h=-4.9(t-9)^2+170 h=height above ground in metres and t=time in seconds

what are the co-ordinates of the vertex of this parabola?

h = (t-9)^2 + 170

h is least when t = 9s
The value of h there is 170.

The vertex is therefore at
t = 9; h=170

your equation

h=-4.9(t-9)^2+170

is in the standard form of the "vertex" form of a quadratic function,
the vertex is (9, 170)

[standard form ...
if y = a(x-h)^2 + k , the vertex is (h,k) ]

To find the coordinates of the vertex of the parabola represented by the relation h = -4.9(t - 9)^2 + 170, we can compare it to the standard form of a quadratic equation: y = a(x - h)^2 + k.

In this case, h represents the horizontal shift, and k represents the vertical shift of the vertex. The coordinates of the vertex are (h, k).

Comparing the given equation to the standard form, we can see that a = -4.9, h = 9, and k = 170.

Therefore, the coordinates of the vertex of this parabola are (9, 170).

To find the coordinates of the vertex of a parabola in the form y = ax^2 + bx + c, you can use the formula:

x = -b / (2a)

In the given relation h = -4.9(t-9)^2 + 170, we can rewrite it as:

h = -4.9t^2 + 88.2t + 43.21

Now we can identify the values of a, b, and c. In this case, a = -4.9, b = 88.2, and c = 43.21.

Using the formula for the x-coordinate of the vertex, we can substitute these values:

x = -88.2 / (2 * -4.9)

Simplifying this equation, we get:

x = -88.2 / -9.8

x = 9

Now, to find the y-coordinate of the vertex, we substitute this value back into the original equation:

h = -4.9(9-9)^2 + 170

Since (9-9)^2 is equal to 0, this eliminates the term involving t. Thus, we are left with:

h = 170

Therefore, the coordinates of the vertex of the parabola are (9, 170).