Exam 3 Review: Precalculus Study Guidance (Polar and Elliptic Equations)

Q31. The graph of a polar equation is given. Select the polar equation for the graph.

Background

Topic: Polar Equations and Graphs

This question tests your ability to recognize and match polar equations to their graphs, specifically those with rose curves (petal-like shapes).

Key Terms and Formulas:

• Polar Equation: An equation where points are defined by radius $r$ and angle $\theta$.

• Rose Curve: A polar graph of the form $r = a \sin(n\theta)$ or $r = a \cos(n\theta)$, where $n$ determines the number of petals.

• Number of Petals: If $n$ is even, the rose has $2n$ petals; if $n$ is odd, it has $n$ petals.

Step-by-Step Guidance

1. Count the number of petals in the graph. This will help determine the value of $n$ in the equation $r = a \sin(n\theta)$ or $r = a \cos(n\theta)$.

2. Check the orientation of the petals. If the petals are aligned with the axes, the equation likely uses cosine; if they are rotated, it likely uses sine.

3. Compare the possible answer choices to the graph. For example, $r = 4 \cos(4\theta)$ produces 8 petals, and $r = 4 \sin(4\theta)$ also produces 8 petals, but with different orientation.

4. Match the amplitude ($a$) and the number of petals to the graph to narrow down the correct equation.

Try solving on your own before revealing the answer!

Final Answer: B) r = 4 cos(4θ)

The graph shows 8 petals, which matches the equation $r = 4 \cos(4\theta)$ (since $n=4$ and $n$ is even, there are $2n=8$ petals).

Q34. Find the standard form of the equation of the ellipse and give the location of its foci. Center at (-2, 3)

Background

Topic: Ellipses in Rectangular Coordinates

This question tests your ability to write the equation of an ellipse given its center and to find the foci based on the ellipse's axes.

Key Terms and Formulas:

• Ellipse Standard Form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ where $(h, k)$ is the center.

• Foci: Located at $(h \pm c, k)$ or $(h, k \pm c)$ depending on the orientation, where $c = \sqrt{a^2 - b^2}$ for $a > b$.

Step-by-Step Guidance

1. Identify the center of the ellipse: $(h, k) = (-2, 3)$.

2. Determine the values of $a$ and $b$ (the lengths of the semi-major and semi-minor axes) from the graph or given information.

3. Write the standard form of the ellipse using the identified values.

4. Calculate $c = \sqrt{a^2 - b^2}$ to find the distance from the center to each focus.

Try solving on your own before revealing the answer!

Final Answer: $(x + 2)^2/36 + (y - 3)^2/9 = 1$; foci at $(-2 + 3\sqrt{3}, 3)$ and $(-2 - 3\sqrt{3}, 3)$

The ellipse is centered at $(-2, 3)$, with $a^2 = 36$ and $b^2 = 9$. The foci are found using $c = \sqrt{36 - 9} = 3\sqrt{3}$, so the foci are $(-2 \pm 3\sqrt{3}, 3)$.