Problem 39 - Algebra 2

Question

Write an equation for each translation. \(y=\cos x, 3\) units to the left and \(\pi\) units up

Step-by-Step Solution

Answer

The final equation after translating \(y=\cos x\) 3 units to the left and \(\pi\) units up is \(y=\cos(x+3)+\pi\)

Step 1: Translate Horizontally

To translate a function horizontally (left or right), add or subtract from the x variable inside the function. In this case, moving 3 units to the left means the 'x' in the cosine function is replaced by \(x+3\). So, \(y=\cos x\) becomes \(y=\cos(x+3)\)

Step 2: Translate Vertically

To translate a function vertically (up or down), add or subtract directly to the function as a whole. Here, moving \(\pi\) units up translates to adding \(\pi\) to the entire function. So, \(y=\cos(x+3)\) now becomes \(y=\cos(x+3)+\pi\)

Step 3: Write the Final Equation

After both translations, the equation of the function becomes \(y=\cos(x+3)+\pi\)

Key Concepts

Horizontal Translation

When you hear "horizontal translation", think about shifting a graph left or right along the x-axis. This type of graph transformation affects the input of the function, which in our exercise is the cosine function, denoted as \(y = \cos x\). Here's how you do it in practice:

A horizontal leftward translation involves adding a positive number to the x-variable. For example, to shift the graph 3 units to the left, replace \(x\) with \(x + 3\). The transformed equation becomes \(y = \cos(x + 3)\).

Remember:

• Add to \(x\) for translation left: More positive \(x\), more steps left!
• Subtract from \(x\) for translation right.

Notice that the function inside does not change its form, just its starting position along the x-axis!

Vertical Translation

Vertical translation moves a graph up and down along the y-axis. This type of transformation impacts the output of the function. With the cosine function, \(y = \cos(x+3)\), you'll move the entire graph up or down by modifying the y-variable.

For example, moving the graph \(\pi\) units upwards involves adding \(\pi\) to the function. So, the equation \(y = \cos(x+3)\) changes to \(y = \cos(x+3) + \pi\). Simple, right?

Key tips:

• Add to the function for upward movement: \(+y\), shifts up!
• Subtract from the function for downward movement.

This kind of vertical translation keeps the shape of the graph intact while repositioning it on the graph paper.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics because they relate angles to distances. The most commonly used trig functions include sine, cosine, and tangent, which are periodic and repetitive in nature. They help describe oscillating systems, such as sound waves or tides.

These functions:

• Sine (\(\sin\))
• Cosine (\(\cos\))
• Tangent (\(\tan\))

create smooth, wave-like graphs. They are defined using the ratios of sides in right-angle triangles or take values based on unit circles. Understanding these can give insights into many real-world phenomena!

Cosine Function

The cosine function specifically deals with the length of the adjacent side of right triangles compared to the hypotenuse. In its graphical form, the cosine function, \(y = \cos x\), produces a wave that starts at its maximum value and oscillates between -1 and 1.

Key characteristics of the cosine wave to remember:

• Period: The length of one complete cycle of the wave is \(2\pi\).
• Amplitude: The height from the center to the peak is 1.
• Symmetry: The cosine curve is even, reflected over the y-axis.

These features show how the cosine function repeats its values in a regular manner. Altering its position or shape is accomplished through transformations, such as horizontal and vertical translations, making it versatile in representing various periodic phenomena.