AP Precalculus Section 3.14 Example: Graph Polar Equation r = 1 +3sinθ by Drawing a Trig Graph

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

To graph the polar equation \( r = 1 + 3\sin(\theta) \) in radians, you can create a table of values for \( \theta \) and calculate the corresponding values of \( r \) using the equation. Here's an example table to help you visualize the graph:

\[
\begin{array}{|c|c|}
\hline
\theta & r \\
\hline
0 & 1 + 3\sin(0) = 1 \\
\frac{\pi}{6} & 1 + 3\sin\left(\frac{\pi}{6}\right) = 1 + \frac{3}{2} = \frac{5}{2} \\
\frac{\pi}{3} & 1 + 3\sin\left(\frac{\pi}{3}\right) = 1 + \frac{3\sqrt{3}}{2} \\
\frac{\pi}{2} & 1 + 3\sin\left(\frac{\pi}{2}\right) = 4 \\
\frac{2\pi}{3} & 1 + 3\sin\left(\frac{2\pi}{3}\right) = 1 - \frac{3\sqrt{3}}{2} \\
\pi & 1 + 3\sin(\pi) = 1 \\
\hline
\end{array}
\]

In this table, we've chosen values of \( \theta \) at \(\frac{\pi}{6}\) rad intervals for simplicity. You can extend the table to cover the range of \( \theta \) values you're interested in. The corresponding values of \( r \) are calculated by substituting each \( \theta \) value into the polar equation.

Once you have the table, plot the points on polar coordinates with \( \theta \) on the angle axis and \( r \) on the radial axis. Connect the points smoothly, and you'll get the graph of the polar equation \( r = 1 + 3\sin(\theta) \) in radians.

The graph is expected to exhibit symmetry due to the sine function. It will oscillate between \(1 - 3\) and \(1 + 3\) as \( \theta \) varies. The amplitude of the oscillation is 3, and the vertical shift is 1.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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