Problem 58 - Algebra and Trigonometry

Question

Graph \(f\) and \(g\) on the same axes, and find their points of intersection. $$f(x)=\sin 2 x, g(x)=2 \sin 2 x+1$$

Step-by-Step Solution

Answer

The functions intersect at points where \(x = \frac{3\pi}{4} + n\pi\), \(n\in\mathbb{Z}\).

Step 1: Understand the function definitions

First, let's identify the two functions we need to work with. The function \(f(x)\) is defined as \(\sin 2x\), and the function \(g(x)\) is defined as \(2 \sin 2x + 1\). These functions involve transformations of the sine function: \(f(x)\) is a horizontally scaled version of the sine function, while \(g(x)\) is both vertically scaled and translated.

Step 2: Identify transformations applied to f(x) and g(x)

For \(f(x) = \sin 2x\), the graph of \(\sin x\) is horizontally compressed by a factor of 2. For \(g(x) = 2 \sin 2x + 1\), the transformation includes compressing the sine function by a factor of 2, vertically stretching it by a factor of 2, and then translating it up by 1 unit.

Step 3: Sketch the transformed functions

Plot the graph of \(f(x) = \sin 2x\) by starting with the base sine wave, compressing it horizontally so that it completes one cycle over \([0, \pi]\). Then, sketch \(g(x) = 2 \sin 2x + 1\); start by doubling the amplitude of the compressed sine wave (amplitude of 2), then shift the entire graph upwards by 1 unit.

Step 4: Find intersection points algebraically

To find the points of intersection algebraically, set \(f(x) = g(x)\). This results in \(\sin 2x = 2 \sin 2x + 1\). Solving this gives \(\sin 2x = -1\). The solutions for \(x\) where \(\sin 2x = -1\) are at \(x = \frac{3\pi}{4} + n\pi\), where \(n\) is an integer, as the sine wave hits -1 at these points.

Step 5: Verify solutions on the graph

Return to the graph to visually confirm that the intersections identified occur where \(\sin 2x = -1\), which are consistent with the points we calculated. Check intervals such as different periods of the graph, observing that both functions align at these key x-values.

Key Concepts

Sine Function

The sine function, denoted as \( \sin x \), is one of the most fundamental trigonometric functions. It is a periodic function, meaning it repeats its values in regular intervals or periods. The standard sine function has a period of \(2\pi\), during which it completes one full cycle. Within this cycle:

• The sine function starts at 0 at \(x = 0\).
• It reaches 1 at \(x = \frac{\pi}{2}\), returning to 0 at \(x = \pi\).
• The sine reaches -1 at \(x = \frac{3\pi}{2}\), and then comes back to 0 at \(x = 2\pi\).

The curve of the sine function is smooth and wave-like. It is symmetrical about the origin, being an odd function, which means that \( \sin(-x) = -\sin(x) \). Understanding the properties of the sine function is crucial when working with its transformations.

Function Transformation

Function transformation involves changing a function's graph in various ways such as shifting, scaling, or reflecting. These transformations allow us to modify the original function into a new form while retaining the sine function’s characteristic periodic shape.For example, in the function \(f(x) = \sin 2x\), the graph is horizontally compressed. This is because the sine function's period is halved to \(\pi\), enabling it to complete two cycles in the usual \(0\) to \(2\pi\) range.In contrast, \(g(x) = 2\sin 2x + 1\) undergoes multiple transformations:

• A horizontal compression by a factor of 2, just like \(f(x)\).
• A vertical stretch since the amplitude is doubled to 2. This means peaks and troughs are twice as high or low as those of \(\sin 2x\).
• An upward shift by 1 unit. This vertical translation moves the entire sine wave one unit higher on the graph.

These transformations are crucial for graphing and comparing functions.

Graphing Trigonometric Functions

Graphing trigonometric functions starts with understanding their base forms. For the sine function, the graph is a smooth curve that repeats every \(2\pi\). Adjustments to amplitude, frequency, and phase shifts define its transformed versions.To graph \(f(x) = \sin 2x\) and \(g(x) = 2\sin 2x + 1\) on the same axes, follow these steps:

• Start with a normal sine graph within one period \([0, 2\pi]\).
• For \(f(x)\), compress the graph so that it completes its cycle faster, precisely over \([0, \pi]\).
• For \(g(x)\), stretch the graph vertically, then shift it upwards by 1. This means its maximum and minimum values are higher than those of \(f(x)\).

Tracking these components allows accurate graph sketching, helping visualize how transformations affect the standard sine wave.

Intersection of Functions

The intersection of functions on a graph represents where two functions have the same value for the same input. Mathematically, this means finding roots of the equation \(f(x) = g(x)\).For the functions \(f(x) = \sin 2x\) and \(g(x) = 2\sin 2x + 1\), setting them equal yields \(\sin 2x = 2\sin 2x + 1\). Simplifying this equation helps discover:

• The requirement \(\sin 2x = -1\) indicates the sine value at these intersections.
• This happens at \(x = \frac{3\pi}{4} + n\pi\), where \(n\) is any integer.

In practice, checking these intervals on the graph assures that the computed mathematical solution matches visual observations, ensuring consistency between algebraic and graphical methods.