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We are asked to determine the quadratic function \( f(x) \) whose graph has the vertex \( (3, -3) \) and passes through the point \( (1, 5) \).

### Step 1: Use the vertex form of a quadratic function.
The vertex form of a quadratic function is:
\[
f(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola. Given that the vertex is \( (3, -3) \), we can substitute \( h = 3 \) and \( k = -3 \), giving us:
\[
f(x) = a(x - 3)^2 - 3
\]
We now need to determine the value of \( a \). To do this, we use the fact that the parabola passes through the point \( (1, 5) \).

### Step 2: Substitute the point \( (1, 5) \) into the equation.
Substitute \( x = 1 \) and \( f(x) = 5 \) into the equation to find \( a \):
\[
5 = a(1 - 3)^2 - 3
\]
\[
5 = a(-2)^2 - 3
\]
\[
5 = 4a - 3
\]
\[
5 + 3 = 4a
\]
\[
8 = 4a
\]
\[
a = 2
\]

### Step 3: Write the final quadratic equation.
Now that we know \( a = 2 \), we can substitute it back into the equation:
\[
f(x) = 2(x - 3)^2 - 3
\]

This is the quadratic function whose graph has the vertex \( (3, -3) \) and passes through the point \( (1, 5) \).

Would you like more details, or do you have any questions?

### Related questions:
1. How can we find the axis of symmetry from the quadratic function?
2. What is the significance of the value of \( a \) in the quadratic function?
3. How do we convert this quadratic function to standard form?
4. How can we find the x-intercepts of the quadratic function?
5. How do we calculate the y-intercept of this quadratic function?

### Tip:
Always check if the quadratic function opens upwards or downwards by inspecting the sign of \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Determine the quadratic function f whose graph is given. The vertex is (3, -3) and the other given point is (1,5).

Learn how to find a quadratic function using the vertex form of the equation, given the vertex (3, -3) and point (1, 5). This problem is suitable for students in Grades 9-11 and covers key algebraic concepts including the vertex form of a quadratic equation.

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