Problem 36 - Algebra and Trigonometry

Question

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ \begin{array}{l}{f(x)=\sin 2 x+1, g(x)=2 \sin 2 x+1} \\ {[-2 \pi, 2 \pi] \text { by }[-1.5,3.5]}\end{array} $$

Step-by-Step Solution

Answer

Intersection points (algebraic): \( x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).

Step 1: Understand the Functions

We need to graph and find intersections for two functions. They are: \( f(x) = \sin 2x + 1 \) and \( g(x) = 2 \sin 2x + 1 \). The functions involve sine waves that are transformed.

Step 2: Graph the Functions

Use a graphing calculator or software to plot the functions \( f(x) = \sin 2x + 1 \) and \( g(x) = 2 \sin 2x + 1 \) over the domain \([-2\pi, 2\pi]\) using the viewing window \([-2\pi, 2\pi]\) for x-values and \([-1.5, 3.5]\) for y-values. Note where the graphs intersect.

Step 3: Find Intersection Points Graphically

By examining the graph, identify the x-values where the lines of the graphs intersect. These will be the approximate solutions, rounded to two decimal places. Through plotting, common points where both functions meet can be determined.

Step 4: Set the Functions Equal Algebraically

To find the intersection points algebraically, set the equations for \( f(x) \) and \( g(x) \) equal to each other: \( \sin 2x + 1 = 2 \sin 2x + 1 \). Simplify this equation to solve for \( x \).

Step 5: Simplify the Equation

Subtract \( \sin 2x + 1 \) from both sides of the equation: \( 0 = \sin 2x \). This implies \( \sin 2x = 0 \).

Step 6: Solve for x

Solve \( \sin 2x = 0 \). This occurs at \( 2x = n\pi \), where \( n \) is an integer. Simplify to get \( x = \frac{n\pi}{2} \).

Step 7: Determine x within the Interval

Determine which values of \( x = \frac{n\pi}{2} \) lie within the interval \([-2\pi, 2\pi]\). The values are \( x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).

Step 8: Verify the Solutions

These x-values should be checked graphically and algebraically by substituting back into both original functions to confirm they intersect at these points.

Key Concepts

Graphing Functions

Graphing functions is a way to visualize mathematical equations so you can better understand their behavior over a particular domain. When dealing with trigonometric functions, such as sine functions, it's essential to understand how these graphs look.
For example, the function \(f(x) = \sin(2x) + 1\) takes the familiar sine wave and shifts it up by one unit. Similarly, the second function \(g(x) = 2\sin(2x) + 1\) not only shifts the sine wave up by one unit but also stretches it vertically by a factor of 2.

• To graph these functions, you can use graphing software or a calculator, which allows you to input the function and automatically generate its graph.
• You should set your viewing window appropriately. Here, the x-values range from \([-2\pi, 2\pi]\) and the y-values from \([-1.5, 3.5]\).
• Observing the graph, you can gather insights on how often the sine functions intersect within the given interval.

Understanding graphing this way helps you visually spot intersections and analyze the periodic nature of trigonometric functions.

Intersection of Functions

The intersection of functions involves finding points where two or more graphs cross each other. These points are solutions to equations where the outputs (y-values) of each function are equal for the same input (x-value).
Identifying these intersections graphically involves:

• Plotting the graphs of both functions and looking for points where their plots converge or cross each other.
• Once intersections are located, the graphical method provides an approximate x-value for the intersection point.

In this problem, you would note where \(f(x) = \sin(2x) + 1\) and \(g(x) = 2\sin(2x) + 1\) intersect visually. Once approximate points are estimated, you can then refine these points algebraically for exact solutions.

Sine Function

The sine function is a fundamental part of trigonometry and has a periodic wave-like pattern. In its simplest form, \(y = \sin(x)\), the graph oscillates between -1 and 1.

• Transformations can shift the graph vertically, horizontally, stretch it, or compress it. For instance, \(\sin(2x)\) represents a sine wave that completes its cycles more quickly than \(\sin(x)\).
• Addition of a constant, such as in \(y = \sin(2x) + 1\), shifts the graph upward.
• When analyzing or graphing these functions, focus on their amplitude (height), period (length of one cycle), and any transformations applied.

This knowledge aids in predicting how various sine functions will look and behave, especially when finding intersections or solutions.

Algebraic Solution

Finding an algebraic solution to the intersection of functions involves setting the equations equal to one another and solving for x. This process provides exact solutions compared to the approximate methods carried out graphically.

• For \(f(x) = \sin(2x) + 1\) and \(g(x) = 2\sin(2x) + 1\), set them equal: \(\sin(2x) + 1 = 2\sin(2x) + 1\).
• By simplifying \(0 = \sin(2x)\), you determine that the sine function equals zero at \(2x = n\pi\), where \(n\) is an integer.
• Solve for x to get \(x = \frac{n\pi}{2}\), then apply the constraint from the domain \([-2\pi, 2\pi]\).
• This yields solutions such as \(-2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\).

These steps not only yield precise x-values for intersections within the specific interval but also reinforce understanding of algebraic manipulation and trigonometric properties.