How to graph a semicircle?
Understand the Problem
The question is asking for guidance on how to create a graph of a semicircle, which involves plotting the points that define its shape based on a given radius and center.
Answer
To graph a semicircle: use the equation $y = k + \sqrt{r^2 - (x - h)^2}$ for $x$ from $h - r$ to $h + r$.
Answer for screen readers
To create a graph of a semicircle, first establish the center $(h, k)$ and radius $r$. Use the equation of the semicircle:
$$ y = k + \sqrt{r^2 - (x - h)^2} $$
for $x$ values from $h - r$ to $h + r$.
Steps to Solve
- Identify the Center and Radius
First, determine the center of the semicircle, which you can denote as $(h, k)$, and the radius $r$.
- Understand the Equation of a Semicircle
If the semicircle opens upward, use the equation: $$ (x - h)^2 + (y - k)^2 = r^2 $$ To get the top half, solve for $y$: $$ y = k + \sqrt{r^2 - (x - h)^2} $$
- Determine the Domain of x-values
The x-values for the semicircle will range from $h - r$ to $h + r$ since that's where the semicircle touches the horizontal axis (assuming it's centered at $(h, k)$).
- Generate Points
Choose x-values within the determined domain and calculate the corresponding y-values using the semicircle equation. For example, if $h = 0$, $k = 0$, and $r = 5$, you can find points by plugging in various $x$-values between -5 and 5.
- Plot the Points on a Graph
Using graph paper or digital graphing tools, plot the points you've calculated. Connect them smoothly to form the shape of a semicircle.
To create a graph of a semicircle, first establish the center $(h, k)$ and radius $r$. Use the equation of the semicircle:
$$ y = k + \sqrt{r^2 - (x - h)^2} $$
for $x$ values from $h - r$ to $h + r$.
More Information
Graphing a semicircle is a great way to visualize the concept of quadratic functions. The semicircle represents half of a circle's equation, commonly used in many fields, including physics and engineering.
Tips
- Using the wrong equation: Confusing the semicircle with the full circle equation.
- Wrong domain selection: Selecting x-values outside the range of $h - r$ and $h + r$.
- Not plotting enough points: Fewer points may lead to a jagged, inaccurate semicircle.
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