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To write the equation of the parabola in vertex form, we start with the general formula of a parabola in vertex form:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.

### Step 1: Identify the vertex
Given that the vertex is \((2, 4)\), we have:
\[
h = 2 \quad \text{and} \quad k = 4
\]
Thus, the equation becomes:
\[
f(x) = a(x - 2)^2 + 4
\]

### Step 2: Use the additional point to find \(a\)
We are given another point \((1, -1)\) that lies on the parabola. Substitute \(x = 1\) and \(f(x) = -1\) into the equation to solve for \(a\):
\[
-1 = a(1 - 2)^2 + 4
\]
\[
-1 = a(1)^2 + 4
\]
\[
-1 = a + 4
\]
\[
a = -1 - 4
\]
\[
a = -5
\]

### Step 3: Write the final equation
Substituting \(a = -5\) back into the equation, we have:
\[
f(x) = -5(x - 2)^2 + 4
\]

### The equation of the parabola in vertex form is:
\[
f(x) = -5(x - 2)^2 + 4
\]

Would you like me to explain any part in more detail, or do you have any questions?

### Here are 5 related questions to expand your understanding:
1. How do you convert a quadratic equation from vertex form to standard form?
2. What is the process to find the vertex of a parabola given in standard form?
3. How do you determine the direction of a parabola (opening upwards or downwards) from the vertex form?
4. How can you find the axis of symmetry from the vertex form equation?
5. What does the value of \(a\) in the vertex form equation tell you about the parabola's width?

### Tip:
When the coefficient \(a\) in the vertex form \(f(x) = a(x - h)^2 + k\) is negative, the parabola opens downwards; if \(a\) is positive, it opens upwards.

Write the equation of the parabola in vertex form. Vertex (2, 4), point (1, -1)

Learn how to write the equation of a parabola in vertex form, given a vertex at (2, 4) and a point (1, -1) on the parabola. This step-by-step solution includes using the vertex form formula and solving for 'a'. Ideal for high school students.

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