determine the focus on each equation(y-2)^2=-4(x-1)

The given equation \((y - 2)^2 = -4(x - 1)\) represents a parabola. To determine the focus of this parabola, we need to rewrite it in the standard form and identify its properties.

### Step 1: Identify the Standard Form
The standard form of a parabola that opens left or right is:

\[
(y - k)^2 = 4p(x - h)
\]

where:
- \((h, k)\) is the vertex of the parabola.
- \(p\) is the distance from the vertex to the focus (if \(p > 0\), the parabola opens to the right; if \(p < 0\), it opens to the left).

### Step 2: Rewrite the Given Equation in Standard Form
The given equation is already in a similar form:

\[
(y - 2)^2 = -4(x - 1)
\]

Here, comparing it with \((y - k)^2 = 4p(x - h)\), we identify:
- \(h = 1\), \(k = 2\) (so the vertex is \((1, 2)\)).
- \(4p = -4\), hence \(p = -1\).

### Step 3: Determine the Focus
Since \(p = -1\), the parabola opens to the left. The focus of a parabola is located \(p\) units away from the vertex along the axis of symmetry.

For this equation:
- The axis of symmetry is horizontal (parallel to the x-axis).
- Since \(p = -1\), the focus is 1 unit left of the vertex \((1, 2)\).

Thus, the focus is:

\[
(h + p, k) = (1 - 1, 2) = (0, 2)
\]

### Final Answer
The focus of the parabola \((y - 2)^2 = -4(x - 1)\) is at the point \(\boxed{(0, 2)}\).

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Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you find the directrix of a parabola?
2. What is the difference between parabolas opening up/down vs. left/right?
3. How do you determine the length of the latus rectum in a parabola?
4. What are the properties of the focus and directrix of a parabola?
5. How can you graph a parabola given its standard form?

**Tip:** For parabolas, remember that the direction of opening is determined by the variable that is squared and the sign of the coefficient in the equation!

determine the focus on each equation (y-2)^2=-4(x-1)

Learn how to determine the focus of a parabola with the equation (y-2)^2 = -4(x-1). This detailed solution explains how to rewrite the equation in standard form, identify the vertex, and calculate the focus. Suitable for students in Grades 10-12.

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