Question

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. \(f(x)=\cos 2 \pi x\), translation 4 units down and 3 units left

Short answer

The rule for \(g\) that represents the indicated transformations of the graph of \(f(x) = \cos(2\pi x)\) is \(g(x) = \cos(2\pi(x + 3)) - 4\).

Step-by-step solution

Original Function

First, start with the original function, which is \(f(x) = \cos(2\pi x)\). This function is the cosine function, but its period is adjusted by the multiplier 2\pi.

Vertical translation

Next, perform the vertical translation. When we say, '4 units down', we subtract 4 from the function. This gives us \(f(x) - 4 = \cos(2\pi x) - 4\).

Horizontal translation

Then, perform the horizontal translation. '3 units to the left' translates into replacing x in the function with \(x + 3\). This leads to \(f(x + 3) - 4 = \cos(2\pi (x + 3)) - 4\).

The Transformed Function

Applying steps 2 and 3, the fully transformed function is thus, \(g(x) = \cos(2\pi (x + 3)) - 4\). This form correctly represents the graph of \(f(x)\) after it has been translated 4 units downwards and 3 units to the left

Key concepts

Cosine Function

The cosine function, represented as \(\cos x\), is a fundamental trigonometric function that plays a vital role in various areas of mathematics, from geometry to calculus. In its most basic form, it calculates the x-coordinate of a point on the unit circle associated with a given angle. For instance, \(\cos (\pi/2)\) equals 0 because the x-coordinate of the point on the unit circle 90 degrees from the positive x-axis is 0.

The function \(f(x) = \cos(2\pi x)\) from our exercise is a cosine wave with some adjustments. Normally, the cosine function has a period of \(2\pi\), which means it completes a full cycle every \(2\pi\) units. However, when a coefficient is applied to the x variable, it affects this period. The \(2\pi\) in front of the x compresses the wave, causing it to complete a cycle more frequently. Specifically, the function \(\cos(2\pi x)\) completes a cycle every 1 unit of x.

In the context of graphing, this function produces a wave-like curve that oscillates above and below the x-axis, intersecting at specific points that represent key features such as maxima, minima, and points of inflection.

Vertical Translation

Vertical translation of a function involves shifting the graph up or down along the y-axis. This transformation is achieved by adding or subtracting a constant value. In the context of our cosine function, the exercise specifies a translation 4 units down. Mathematically, this is represented by subtracting 4 from the original function, leading to \(f(x) - 4\).

To visualize this, imagine the entire graph of the cosine function being moved 4 units lower. All the points on the original graph are now 4 units closer to the negative y-axis. The prominent features like crests and troughs of the wave maintain their shape but are repositioned vertically. It's important to note that this translation does not affect the period or the frequency of the wave; it simply changes its vertical location.

Horizontal Translation

Horizontal translation shifts the graph of a function to the left or right along the x-axis. This is accomplished by replacing every occurrence of x in the original function with \(x+c\) or \(x-c\), where c is a constant. Translating a graph to the left by 3 units means we use \(x+3\), incorporating this within our function leading to \(f(x+3)\).

In the given exercise, this modification results in \(\cos(2\pi (x + 3))\). The plus sign in this case indicates a shift to the left. This type of horizontal shift does not alter the height or the period of the cosine wave; it changes the starting point of the wave. After shifting, the points where the function originally intersected the x-axis are now moved 3 units to the left. This transformation can sometimes be counterintuitive since the function \(f(x+3)\) indicates a leftward shift, even though the number inside the parentheses is positive.