Figure 1
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.
Figure 2
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.
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.
Figure 4
Images/Mathematical drawings are created with GeoGebra.