The problem provides the graph of a quadratic function $f(x) = x^2 + bx + c$ and two points on the graph: $(0, -2)$ and $(1, -3)$. The task is to find the values of $b$ and $c$, and then find the value of $y$ if $(2, y)$ is a point on the graph.

1. Problem Description

The problem provides the graph of a quadratic function $f(x) = x^2 + bx + c$ and two points on the graph: $(0, -2)$ and $(1, -3)$. The task is to find the values of $b$ and $c$, and then find the value of $y$ if $(2, y)$ is a point on the graph.

2. Solution Steps

(i) Find the values of $b$ and $c$.

Since $(0, -2)$ is a point on the graph, $f(0) = -2$. Substituting $x=0$ into the equation gives:

$f(0) = 0^2 + b(0) + c = c$

So, $c = -2$.

Now the equation is $f(x) = x^2 + bx - 2$.

Since $(1, -3)$ is a point on the graph, $f(1) = -3$. Substituting $x=1$ into the equation gives:

$f(1) = 1^2 + b(1) - 2 = -3$

$1 + b - 2 = -3$

$b - 1 = -3$

$b = -3 + 1 = -2$

So, $b = -2$.

Therefore, $f(x) = x^2 - 2x - 2$.

(ii) If $(2, y)$ is a point on the graph, find the value of $y$.

Since $(2, y)$ is a point on the graph, $f(2) = y$. Substituting $x=2$ into the equation gives:

$y = f(2) = 2^2 - 2(2) - 2$

$y = 4 - 4 - 2$

$y = -2$

3. Final Answer

(i) $b = -2$ and $c = -2$

(ii) $y = -2$