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The image presents a problem based on a quadratic function of the form \( f(x) = ax^2 + bx + c \). It provides information about the vertex and y-intercept of the quadratic and asks for:

### (a) 
(i) **Equation of the axis of symmetry**  
(ii) **The value of \( c \)**  

### (b) 
Finding the values of \( a \) and \( b \).

Let's work through the solution step by step.

### Information provided:
- Vertex: \( A(1, 7) \)
- Point on the y-axis: \( B(0, 2) \)
  
#### (a)(i) Equation of the Axis of Symmetry:
The axis of symmetry of a parabola is the vertical line that passes through the x-coordinate of the vertex. Since the vertex is at \( (1, 7) \), the equation of the axis of symmetry is:
\[
x = 1
\]

#### (a)(ii) Value of \( c \):
The point \( B(0, 2) \) is where the graph crosses the y-axis. This means \( f(0) = 2 \), and since the y-intercept of the quadratic function \( f(x) = ax^2 + bx + c \) occurs when \( x = 0 \), the value of \( c \) is simply the y-coordinate of point \( B \), i.e.:
\[
c = 2
\]

#### (b) Finding the values of \( a \) and \( b \):

We can use the vertex form of a quadratic function, which is given by:
\[
f(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola. Given the vertex is \( (1, 7) \), the equation becomes:
\[
f(x) = a(x - 1)^2 + 7
\]
We also know that the point \( B(0, 2) \) lies on the parabola, so substituting \( x = 0 \) and \( f(0) = 2 \) into the equation gives:
\[
2 = a(0 - 1)^2 + 7
\]
\[
2 = a(1) + 7
\]
\[
a = 2 - 7 = -5
\]
Thus, \( a = -5 \).

Now that we know \( a = -5 \), we can expand the vertex form equation back into the standard quadratic form:
\[
f(x) = -5(x - 1)^2 + 7
\]
Expanding this:
\[
f(x) = -5(x^2 - 2x + 1) + 7
\]
\[
f(x) = -5x^2 + 10x - 5 + 7
\]
\[
f(x) = -5x^2 + 10x + 2
\]
From this, we see that \( b = 10 \).

### Final Answers:
- (a)(i) The equation of the axis of symmetry is \( x = 1 \).
- (a)(ii) The value of \( c \) is \( 2 \).
- (b) The value of \( a \) is \( -5 \) and the value of \( b \) is \( 10 \).

Let me know if you want more details or have any questions!  
Here are some related problems to consider:
1. What is the effect of changing the value of \( a \) on the graph?
2. How do you find the x-intercepts of a quadratic function?
3. What is the significance of the vertex in a quadratic function?
4. How would you find the minimum or maximum value of a quadratic function?
5. What is the discriminant, and how does it relate to the number of roots of a quadratic equation?

**Tip:** When solving quadratic problems, always identify key points like the vertex and intercepts, as they provide crucial information about the function's shape and direction.

The graph of the quadratic function f(x) = ax^2 + bx + c has its vertex at the point A (1, 7) and crosses the y-axis at the point B (0, 2). (a) Write down: (i) the equation of the axis of symmetry. (ii) the value of c. (b) Find the value of a and the value of b.

Solve a quadratic function problem involving the vertex at point A (1, 7) and a y-intercept at B (0, 2). Learn how to find the axis of symmetry, value of c, and calculate the values of a and b. Suitable for Grades 9-11 students.

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