The problem provides an incomplete table of values for the function $y = 3 - (x-2)^2$. The task involves completing the table using the symmetry of the graph, plotting the graph, finding the maximum value, finding the equation of the axis of symmetry, finding the roots of the equation $3 - (x-2)^2 = 0$, and determining the equation of a function with minimum value -3 and axis of symmetry $x = 2$.

1. Problem Description

The problem provides an incomplete table of values for the function $y = 3 - (x-2)^2$. The task involves completing the table using the symmetry of the graph, plotting the graph, finding the maximum value, finding the equation of the axis of symmetry, finding the roots of the equation $3 - (x-2)^2 = 0$, and determining the equation of a function with minimum value -3 and axis of symmetry $x = 2$.

2. Solution Steps

(i) Completing the table: The function $y = 3 - (x-2)^2$ is a quadratic function whose graph is a parabola. The axis of symmetry is $x = 2$. Given the x-values -1, 0, 1, 2, 3, 4, 5, the corresponding y-values are -6, 2, 3, 2, -1, -

6. Since the axis of symmetry is $x = 2$, the y-value when $x = 0$ is the same as when $x = 4$, and the y-value when $x = 1$ is the same as when $x = 3$, and the y-value when $x = -1$ is the same as when $x=5$. Thus, when $x=0$, $y=2$.

(ii) Plotting the graph: The graph should be a parabola opening downwards with vertex at $(2, 3)$. Plot the points (-1,-6), (0, 2), (1, 3), (2, 3), (3, 2), (4, -1), (5, -6).

(iii) Maximum value: From the graph and the table, the maximum value of y is

3.

(iv) Equation of the axis of symmetry: The axis of symmetry is $x = 2$.

(v) Finding the roots of $3 - (x-2)^2 = 0$:

$3 - (x-2)^2 = 0$

$(x-2)^2 = 3$

$x-2 = \pm \sqrt{3}$

$x = 2 \pm \sqrt{3}$

So, the roots are $x = 2 + \sqrt{3}$ and $x = 2 - \sqrt{3}$.

(vi) Equation of a function with minimum value -3 and axis of symmetry $x=2$: A parabola with a minimum value has the form $y = a(x-h)^2 + k$ where $a > 0$, $(h, k)$ is the vertex. The axis of symmetry is $x = h = 2$. The minimum value is $k = -3$. The simplest equation is when $a = 1$, so $y = (x-2)^2 - 3$.

3. Final Answer

(i) y = 2 when x = 0

(iii) Maximum value = 3

(iv) Equation of the axis of symmetry: x = 2

(v) Roots: $x = 2 + \sqrt{3}$ and $x = 2 - \sqrt{3}$

(vi) Equation of the function: $y = (x-2)^2 - 3$