Problem 38 - Algebra 1: Concepts and Skills

Question

Graph the exponential function. $$y=\left(\frac{1}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(y=(\frac{1}{2})^{x}\) is a continuous, decreasing curve. It decreases from the upper left, crossing the y-axis at \(y = 1\) (when \(x = 0\)), and continues to decrease but never reaches \(y = 0\) (the x-axis), no matter how far extended to the right.

Step 1: Choose the X-values

X-values typically chosen are usually around \(x = -2, -1, 0, 1, 2\). These will provide a clear picture of the graph. Higher and lower x-values can also be considered, but the graph may stretch quickly to the infinities.

Step 2: Compute the Corresponding Y-values

Substitute each x-value into the equation \(y=(\frac{1}{2})^{x}\) and compute the equivalent y-value. For example, substitute \(x=0\), into the equation: \(y=(\frac{1}{2})^{0}\), the resulting value of \(y=1\). Do this for all other x-values as well.

Step 3: Plot the Points and Draw the Graph

Plot the points that are formed from pairs of x and y-values on a graph. Draw a curve through these plotted points to represent the graph of the function \(y=(\frac{1}{2})^{x}\). The graph will be a decreasing curve, as it's an exponential decay function.

Key Concepts

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. This is common in scenarios like radioactive decay, population decline, or cooling processes. The function provided, \( y=\left(\frac{1}{2}\right)^{x} \), is an example of exponential decay because the base of the exponential is less than 1. Each time the value of \( x \) increases by 1, the value of \( y \) is halved.

• This results in a graph that continuously decreases as it moves from left to right along the x-axis.
• The initial value of the function at \( x=0 \) is 1, since any number raised to the power of 0 is 1.
• As \( x \) becomes positive, \( y \) approaches 0 but never quite reaches it, creating a horizontal asymptote at \( y=0 \).
• When \( x \) is negative, the function actually grows towards positive infinity, though this is less intuitive since we often associate decay with declining values.

Understanding this behavior helps in visualizing how rapidly a quantity reduces under exponential conditions, provided the base is fractionally smaller than 1.

Graphing Functions

Graphing functions involves translating an equation into visual form, allowing a clearer understanding of its behavior and properties. For the exponential function \( y=\left(\frac{1}{2}\right)^{x} \), each point on the graph corresponds to a particular \( x \)-value and its associated \( y \)-value.

• Start by choosing specific \( x \)-values, like \(-2, -1, 0, 1,\) and \(2\), to explore what the changes in \( x \) do to \( y \).
• Compute the \( y \)-values by substituting these into the equation. For example, \( x=2 \) gives \( y=\left(\frac{1}{2}\right)^2=\frac{1}{4} \).
• Plotting these \( (x, y) \) pairs on a graph provides the points through which the curve of the function is drawn.
• The resulting curve for \( y=\left(\frac{1}{2}\right)^{x} \) slopes downwards from left to right, clearly illustrating the function's exponential decay.

This graphical representation is vital in identifying key elements like the rate of decline, intercepts, and asymptotes, contributing to a better intuitive grasp of the function's nature.

Algebra 1

Algebra 1 introduces foundational concepts which empower students to tackle more complex mathematical tasks involving equations and functions. With exponential functions, students learn about the behavior of equations with variables in exponents, such as \( y=\left(\frac{1}{2}\right)^{x} \).

• The function highlights the importance of understanding base and exponent relationships, crucial for predicting changes in growth or decay.
• Students use substitution to evaluate the function at various \( x \)-values, consolidating skills in manipulating equations.
• Graphing the function visually demonstrates the concept of decay, with each increase in \( x \) corresponding to an exponential drop in \( y \).
• Critical thinking comes into play in analyzing how different bases (greater or lesser than 1) affect the function’s behavior.

The skills developed in Algebra 1 are not just about solving equations but also about interpreting and predicting real-world phenomena through mathematical models like exponential decay.