The problem provides an incomplete table of values for the function $y = x(x-2)-3$. We need to: (i) Find the value of $y$ when $x=2$. (ii) Draw the graph of the function using a suitable scale on a standard coordinate plane. (iii) Write down the coordinates of the turning point of the graph. (iv) Find the roots of the equation $x^2 - 2x - 3 = 0$ using the graph. (v) Describe the behavior of $y$ when $-1 \le x \le 1$.

1. Problem Description

The problem provides an incomplete table of values for the function $y = x(x-2)-3$. We need to:

(i) Find the value of $y$ when $x=2$.

(ii) Draw the graph of the function using a suitable scale on a standard coordinate plane.

(iii) Write down the coordinates of the turning point of the graph.

(iv) Find the roots of the equation $x^2 - 2x - 3 = 0$ using the graph.

(v) Describe the behavior of $y$ when $-1 \le x \le 1$.

2. Solution Steps

(i) Find the value of $y$ when $x=2$.

We have the function $y = x(x-2) - 3$.

Substitute $x=2$ into the equation:

$y = 2(2-2) - 3 = 2(0) - 3 = 0 - 3 = -3$

(ii) Draw the graph of the function.

We complete the table using $y = x(x-2)-3 = x^2-2x-3$:

When $x=-2$, $y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5$

When $x=-1$, $y = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$

When $x=0$, $y = (0)^2 - 2(0) - 3 = 0 - 0 - 3 = -3$

When $x=1$, $y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$

When $x=2$, $y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3$

When $x=3$, $y = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$

When $x=4$, $y = (4)^2 - 2(4) - 3 = 16 - 8 - 3 = 5$

The completed table is:

x | -2 | -1 | 0 | 1 | 2 | 3 | 4

--|----|----|----|----|----|---|---

y | 5 | 0 | -3 | -4 | -3 | 0 | 5

We would then plot these points on a coordinate plane and draw a smooth curve through them. (Since I cannot draw a graph here, imagine plotting the points and connecting them with a curve. The graph is a parabola.)

(iii) Coordinates of the turning point.

From the completed table or the equation, we can see that the turning point (vertex) of the parabola is at $(1, -4)$.

(iv) Roots of the equation $x^2 - 2x - 3 = 0$.

The roots of the equation are the x-values where the graph intersects the x-axis, i.e., where $y=0$.

From the graph (or the table), we can see that the graph intersects the x-axis at $x=-1$ and $x=3$. Therefore, the roots are $x=-1$ and $x=3$.

(v) Behavior of $y$ when $-1 \le x \le 1$.

When $-1 \le x \le 1$, the y-values decrease from $0$ to $-4$. So, $y$ is decreasing.

3. Final Answer

(i) When $x=2$, $y = -3$.

(ii) Graph: (Cannot draw the graph here)

(iii) Turning point: $(1, -4)$

(iv) Roots: $x = -1$ and $x = 3$

(v) When $-1 \le x \le 1$, $y$ is decreasing.