We want to write the equation of the given parabola. To do so, let's recall the vertex form of a quadratic function.

y = a (x - h)^{2} + k

In this expression,

a,

h,

and

k

are either positive or negative constants. Let's start by identifying the vertex.

The vertex of this parabola has coordinates

(2, - 9) .

This means that we have

h = 2

and

k = - 9 .

We can use these values to partially write our function.

y = a (x - (2))^{2} + (- 9) ⇕ y = a (x - 2)^{2} - 9

We can see in the graph that the parabola opens upwards. Therefore,

a

will be a positive number. To find its value, we will use the given point that is not the vertex.

We can see above that the point has coordinates

(0, - 5) .

Since this point is on the curve, it satisfies its equation. Hence, to find the value of

a,

we can substitute

0

for

x

and

- 5

for

y

and simplify.

y = a (x - 2)^{2} - 9

$x = 0$ , $y = - 5$

- 5 = a (0 - 2)^{2} - 9

▼

Solve for

a

Add terms

- 5 = a (- 2)^{2} - 9

Calculate power

- 5 = a (4) - 9

$LHS + 9 = RHS + 9$

4 = a (4)

$LHS / 4 = RHS / 4$

1 = a

Rearrange equation

a = 1

We found that

a = 1 .

Now we can complete the equation of the curve.

y = 1 (x - 2)^{2} - 9

To obtain the standard form of the quadratic function, we simplify the right-hand side of the equation.

y = (x - 2)^{2} - 9

$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

y = x^{2} - 2 x (2) + 2^{2} - 9

Calculate power and product

y = x^{2} - 4 x + 4 - 9

Subtract term

y = x^{2} - 4 x - 5

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