Find an expression for the function whose graph is the given curve.

The bottom half of the parabola *x* + (*y* − 4)2 = 0

To find the expression for the function whose graph represents the bottom half of the given parabola, we begin with the given equation:

\[
x + (y - 4)^2 = 0
\]

### Step 1: Solve for \(x\) in terms of \(y\)
Rearrange the equation:

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

This equation now expresses \(x\) as a function of \(y\). However, we want to find the expression for \(y\) as a function of \(x\).

### Step 2: Solve for \(y\) in terms of \(x\)
We have:

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

Now, to isolate \(y\), take the square root of both sides:

\[
\sqrt{-x} = y - 4
\]

So,

\[
y = 4 \pm \sqrt{-x}
\]

### Step 3: Choose the bottom half of the parabola
The expression \(y = 4 + \sqrt{-x}\) corresponds to the upper half of the parabola, while \(y = 4 - \sqrt{-x}\) corresponds to the **bottom half**. Thus, the expression for the function representing the bottom half is:

\[
y = 4 - \sqrt{-x}
\]

### Final Answer:
The function for the bottom half of the parabola is:

\[
y = 4 - \sqrt{-x}
\]

### Would you like further clarification or more details?

Here are 5 related questions to explore further:
1. How do we derive the equation of a parabola from its geometric properties?
2. Why does the expression use \(\sqrt{-x}\) instead of \(\sqrt{x}\)?
3. How do we find the vertex of the parabola from the given equation?
4. What changes if we want the equation for the upper half of the parabola?
5. How would we graph this function?

**Tip:** Always be careful when taking square roots. Remember to account for both the positive and negative roots!

Find an expression for the function whose graph is the given curve. The bottom half of the parabola x + (y − 4)^2 = 0

Explore how to derive the function for the bottom half of the parabola defined by x + (y − 4)^2 = 0. This solution involves key algebraic principles, solving quadratic equations, and applying the square root property. Suitable for students in Grades 10-12.

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