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 $(1, 5)$.

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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.