Problem 52 - Algebra 2

Question

How is the graph of \(y=4 \cdot\left(\frac{1}{2}\right)^{x}+3\) translated from the graph of \(y=4 \cdot\left(\frac{1}{2}\right)^{x} ?\) \(\begin{array}{llll}{\text { E. } 3 \text { units right }} & {\text { 6. } 3 \text { units left }} & {\text { H. } 3 \text { units down }} & {\text { 1. } 3 \text { units up }}\end{array}\)

Step-by-Step Solution

Answer

The graph is translated 3 units up.

Step 1: Identify the translation

The original function is \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\) and the modified function is \(y=4 \cdot\left(\frac{1}{2}\right)^{x} + 3\). The change occur in the y-value due to the addition of +3.

Step 2: Determine the direction of translation

The addition of +3 in the equation will move graph 3 units upwards. This is because the +3 is added to whole function, affecting the y-value of all points on the graph. This results in a vertical translation.

Step 3: Conclusion

So, the graph of the function \(y=4 \cdot\left(\frac{1}{2}\right)^{x}+3\) is translated 3 units up from the graph of the function \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable, often denoted as \(x\), appears in the exponent. This means that the function changes at a rate proportional to its current value. An example is \(y = 4 \cdot \left(\frac{1}{2}\right)^{x}\), where \(4\) is the initial value or coefficient, and \(\frac{1}{2}\) is the base of the exponential component. These functions are widely used to model situations of exponential growth or decay, such as population growth or radioactive decay.

Characteristics of exponential functions include:

• A continuous, smooth curve that never touches the x-axis, becoming asymptotic as it approaches zero.
• For bases between zero and one, the function represents exponential decay. Conversely, a base greater than one indicates exponential growth.

Understanding exponential functions is crucial in graph transformations as they set the foundation for further modifications.

Vertical Translation

A vertical translation in graph theory involves shifting a graph up or down along the y-axis. This is often due to a constant being added or subtracted from the entire function. In the problem we've studied, the original function was \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\), and the modified function is \(y=4 \cdot\left(\frac{1}{2}\right)^{x} + 3\).

A vertical translation can be easily identified:

• If there's an addition like "+3," the graph moves upwards by this amount.
• If there's a subtraction, the graph shifts down.

This operation only affects the y-values, leaving the x-values unchanged. The entire graph elevates by the value of the added constant, affecting all points uniformly. Thus, in this case, the graph shifts up 3 units due to the addition of the constant \(+3\).

Graph Transformations

Graph transformations are techniques used to manipulate the original structure of a graph to produce a new graph. These transformations can include translations, reflections, stretches, and compressions. In the context of our exercise, we focused on a graph translation caused by a vertical shift. Translating a graph without altering its shape maintains the curve's properties while changing its position.

Key types of graph transformations include:

• Translations: Moving the graph without rotation. Includes vertical (up/down) and horizontal (left/right) translations.
• Reflections: Flipping the graph over a specific axis, often the x-axis or y-axis.
• Stretches and compressions: Altering the graph's scale vertically or horizontally.

By comprehending these transformations, you can predict and visualize changes to graph structures when equations are modified. Recognizing these patterns is a crucial skill in algebra and calculus.