Problem 51 - Algebra 2

Question

How is the graph of \(y=4 \cdot\left(\frac{1}{2}\right)^{x-3}\) translated from the graph of \(y=4 \cdot\left(\frac{1}{2}\right)^{x} ?\) \(\begin{array}{llll}{\text { A. } 3 \text { units right }} & {\text { B. } 3 \text { units left }} & {\text { C. } 3 \text { units down }} & {\text { D. } 3 \text { units up }}\end{array}\)

Step-by-Step Solution

Answer

A. 3 units right

Step 1: Understand the Original Function

The original function is \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\). This is an exponential decay function which decreases as x increases.

Step 2: Analyze the Transformed Function

The new function is \(y=4 \cdot\left(\frac{1}{2}\right)^{x-3}\). The '-3' inside the exponent means the graph of the function is horizontally shifted by a certain amount.

Step 3: Determine the Direction of Transformation

A positive number subtracted from 'x' in the exponent indicates a shift to the right by that amount. Therefore, the graph of the given equation \(y=4 \cdot\left(\frac{1}{2}\right)^{x-3}\) is the graph of \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\) shifted 3 units to the right.

Key Concepts

Horizontal Shifts

In graphing, a horizontal shift moves the entire graph of a function left or right along the x-axis. For exponential functions, these shifts are indicated by changes in the exponent.

When you see an expression of the form \(y = a \, b^{x - h}\), the \(-h\) inside the exponent tells us how to shift the graph horizontally. A negative \(h\) (as in \(x - h\)) indicates a shift to the right by \(h\) units, while a positive \(h\) (as in \(x + h\)) indicates a shift to the left.

In our problem, the function is \(y = 4 \left( \frac{1}{2} \right)^{x-3}\). The term \(-3\) suggests a horizontal shift to the right by 3 units. This means every point on the original graph moves 3 units right along the x-axis, altering the graph's position but not its shape.

• Horizontal shifts do not affect the range.
• The domain remains all real numbers after the shift.

Exponential Decay

Exponential decay refers to a process where quantities decrease rapidly at first and then slowly over time. The base of an exponential function determines the nature of the decay or growth. In this function, \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\), the base \(\frac{1}{2}\) is between 0 and 1, signaling exponential decay.

As \(x\) increases, the value \(\left(\frac{1}{2}\right)^{x}\) gets smaller, explaining the rapid decrease. This characteristic of exponential decay shapes the graph into a curve that quickly levels off, approaching, but never touching, the x-axis. The rate of decrease depends on the base and the scaling factor (4 in the original function), influencing how rapidly values decline after \(x = 0\).

Exponential decay functions are common in real-world scenarios:

• Radioactive decay in physics.
• Depreciation of asset values in economics.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting a graph to create new functions from existing ones. They allow us to visualize effects of algebraic changes to equations.

For the exponential function \(y=4 \cdot\left(\frac{1}{2}\right)^{x}\), a horizontal shift is one of the simplest transformations. It modifies only the graph's position but maintains the fundamental shape and orientation of the curve.

Imagine stretching a piece of rubber horizontally; the shape of the rubber remains constant, but its location changes. Similarly, horizontal transformations in equations like \(y=4 \cdot\left(\frac{1}{2}\right)^{x-3}\) influence the graph's positioning. When observing a transformed function:

• Assess whether the shift happens horizontally or vertically.
• Check if the factors alter the steepness or direction of the graph.

Understanding graph transformations makes it easier to predict how variations in equations impact their graphical representations.