This question aims to find an **expression** for the **function** whose **graph** is given by the **curve** $x^2 + (y – 4)^2 = 9$. The graph is shown in Figure 1.

This question is based on the concept of **circle geometry** and **basic calculus.** We can find an **expression** of the function from the given curve equation by simply **solving for its output value**. The **curve equation** is given, representing a **circle** shown in Figure 1.

## Expert Answer

The **circle equation,** when solved for $y$, gives two expressions, one **positive** and the other **negative,** due to the **square root.** Those expressions represent the **two halves** of the **same circle.** The **positive expression** shows the **upper semi-circle,** while the **negative** expression shows the **lower semi-circle.**

The equation of the circle is given as:

\[ x^2 + (y – 4)^2 = 9 \]

If we solve this equation’s output, that is, $y$, we can find the **expression** for the **function.**

\[ (y – 4)^2 = 9 – x^2 \]

Taking **square root** on both sides:

\[ \sqrt {(y – 4)^2} = \pm \sqrt {9 – x^2} \]

\[ y – 4 = \pm \sqrt {9 – x^2} \]

\[ y = \pm \sqrt {9 – x^2} + 4 \hspace {0.4in} (1) \]

The equation $(1)$ shows the **two halves** of the **circle.** We take the **positive expression** to show its graph in Figure 2, which is the **upper half of the circle**.

## Numerical Results

The **expression** for the **function** of the given **curve** is solved as:

\[ y = \pm \sqrt {9 – x^2} + 4 \]

We can also write this equation as the **function** of $x$:

\[ f(x) = \pm \sqrt {9 – x^2} + 4 \]

## Alternative Solution

Given the **circle equation,** we can directly solve for $y$.

\[ (x – a)^2 + (y – b)^2 = r \]

\[ y = \pm \sqrt {r – (x – a)^2} + b \]

Using the above equation, we can directly calculate the expression for the function of the **given curve**.

## Example

The **equation** of the **curve** is given as $(x – 4)^2 + y^2 = 25$, which represents a circle. Find the expression for the function.

The equation $(x -4)^2 + y^2 = 25$ represents a circle shown in Figure 3.

Solving the **equation’s output,** we can find the expression for the function.

\[ (x – 4)^2 + y^2 = 25 \]

\[ y^2 = 25 – (x – 4)^2 \]

\[ \sqrt {y^2} = \pm \sqrt {25 – (x – 4)^2} \]

\[ y = \pm \sqrt {25 – (x – 4)^2} \]

We can represent this equation as a **function** of $x$ as:

\[ f(x) = \pm \sqrt {25 – (x – 4)^2} \]

This function represents the **two halves** of the **circles** shown in Figure 3. We only take the **positive expression** to represent its **graph** in Figure 4 below.

*Images/Mathematical drawings are created with GeoGebra.*