Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r=1-\sin \theta$$

Short answer

Question: Graph the polar equation $$r = 1 - \sin\theta$$ and find its vertex. Answer: The graph of the polar equation $$r = 1 - \sin\theta$$ is a downward-opening parabola with its vertex at the point $$\left(\frac{1}{2}, -\frac{1}{4}\right)$$.

Step-by-step solution

Convert to Cartesian Coordinates

To convert the polar equation $$r = 1 - \sin\theta$$ to Cartesian coordinates, we can use the equations $$x = r\cos\theta$$ and $$y = r\sin\theta$$. First, we isolate $$\sin\theta$$ in our polar equation: $$\sin\theta = 1 - r$$ Now substitute $$x = r\cos\theta$$ and $$y = r\sin\theta$$: $$y = r\sin\theta = r(1 - r)$$ Expanding the equation, we get: $$y = r - r^2$$ Now we have a Cartesian equation that we can graph.

Find the Vertex

The equation $$y = r - r^2$$ is a parabola opening downward. We can find its vertex by completing the square. First, rewrite the equation by factoring out a -1 from the r^2 term: $$y = -(-1r^2 + r)$$ Next, we complete the square. Add and subtract 1/4 inside the parentheses: $$y = -(-1r^2 + r + \frac{1}{4} - \frac{1}{4})$$ Now factor the quadratic inside the parentheses: $$y = -(-1(r - \frac{1}{2})^2 - \frac{1}{4})$$ So the vertex of the parabola is at the point $$\left(\frac{1}{2}, -\frac{1}{4}\right)$$.

Graph the Equation

Now we are ready to graph the equation $$y = r - r^2$$. We found its vertex in the previous step. Since it's a downward-opening parabola, we know that the graph will be symmetric about the line $$r = \frac{1}{2}$$. Plot the vertex and several other points, such as $$(0,0), \left(\frac{1}{4}, \frac{3}{16}\right), (1,0), \text{ and } \left(\frac{3}{4}, \frac{3}{16}\right)$$. Sketch a smooth curve through these points to create the graph of the equation.

Check Your Work Using a Graphing Utility

To check your work, input the polar equation $$r = 1 - \sin\theta$$ into a graphing utility with polar coordinate mode. Compare the graphs to see if they match. If they do, congratulations! You have successfully graphed the given polar equation.

Key concepts

Graphing Utility

A graphing utility is a technological tool that helps visualize mathematical equations by generating their graphs. It allows you to explore complex mathematical concepts with ease and precision.
One of its primary functions is to convert mathematical equations into visual representations that help compare and analyze their behavior and properties. When working on problems involving polar coordinates, such as the equation \( r = 1 - \sin\theta \), a graphing utility can be invaluable.

• It helps verify the accuracy of the manually plotted graph.
• You can switch between polar and Cartesian coordinate modes to see different views of the same equation.
• It allows you to plot multiple equations simultaneously for comparison.

By inputting the equation into a graphing utility, you can see firsthand how the parabola is shaped and ensure that your manual graph aligns with the computer-generated one. This process is an excellent way to check your understanding and ensure accuracy.

Cartesian Coordinates

In mathematics, Cartesian coordinates provide an efficient way to describe the position of points in a plane using ordered pairs \((x, y)\). They derive their name from René Descartes, the French mathematician who introduced this coordinate system.
Cartesian coordinates allow you to translate polar equations into a context that's often easier to work with:

• The x-coordinate corresponds to the horizontal position, calculated as \(x = r\cos\theta\).
• The y-coordinate represents the vertical position, given by \(y = r\sin\theta\).

Converting the polar equation \( r = 1 - \sin \theta \) into Cartesian form provides the equation \( y = r - r^2 \). This is essential because many graphing utilities and mathematical analyses require Cartesian coordinates to plot and understand graphs fully. By moving between coordinate systems, you gain a better grasp of the geometric nature of the equation.

Parabola

A parabola is a symmetrical plane curve, which is the graph of a quadratic function. It takes on a U-shaped form, but it can open upwards or downwards depending on the equation.When working with the Cartesian equation \( y = r - r^2 \), it represents a downward-opening parabola:

• It's vertex is at \( \left( \frac{1}{2}, -\frac{1}{4} \right) \).
• This point is the maximum of the parabola because it opens downwards.
• The symmetry of a parabola makes it easier to plot as you only need half of the points, and you can reflect them over the axis of symmetry.

Knowing how to identify the vertex and direction of a parabola is crucial for graphing it accurately. Utilizing graphing utilities together with manual plotting ensures that you understand both the algebra and geometry of the equation.