Problem 33 - Algebra 1: Concepts and Skills

Question

Graph the exponential decay model. $$ y=10\left(\frac{1}{2}\right)^{t} $$

Step-by-Step Solution

Answer

The graph starts at (0,10) and decreases towards the x-axis, but never touches it. The domain of the function is all real numbers, and the range is all positive real numbers.

Step 1: Identify the y-intercept

The y-intercept of an exponential function is generally the basic form \(y=a(b)^x\), where the y-intercept is \(a\). Here, \(a=10\), hence the y-intercept of the function is (0,10).

Step 2: Identify the shape of curve

As \(b=1/2\) which is less than 1, the graph will represent an exponential decay. This means the curve will start at the y-intercept and decrease towards the x-axis, never quite reaching it.

Step 3: Identify the Domain and the Range

For the exponential function, the domain which is the possible values for \(x\) (in our case \(t\)) is all real numbers. The range, which are the possible values for \(y\), is \(y>0\) since the graph never crosses the x-axis.

Step 4: Draw the Graph

Start at the y-intercept identified in Step 1. The graph should then fall towards the x-axis, reflecting the shape identified in Step 2. Note that the graph will never touch the x-axis, matching the range identified in Step 3.

Key Concepts

Exponential Decay Model

When we talk about the exponential decay model, we're looking at a type of function that decreases rapidly initially and then slows down as it approaches zero. In this model, the formula typically looks like y = a(b)^t, where a is the initial value, b is the base representing the decay factor (which is between 0 and 1), and t represents time or another independent variable.

An important characteristic of the exponential decay model is that the quantity never actually reaches zero, which means the decay process continues indefinitely. This is extremely useful in modeling real-world phenomena like radioactive decay, cooling of hot objects, or depreciation of assets over time.

By analyzing the base b, we can determine the rate at which the value is decreasing. In the given exercise y=10(1/2)^t, the decay rate is 50% per time unit, meaning that the quantity is halved with each time increment. Understanding this model is crucial for interpreting data and predictions in various scientific and financial fields.

Y-intercept of Exponential Function

The y-intercept is a significant point in the graph of an exponential function, as it represents the value of y when the independent variable (in our case, t) is zero. For our exercise y=10(1/2)^t, identifying the y-intercept is pretty straightforward: you set the t to zero, and whatever coefficient is in front of the base becomes your y-intercept. Why? Because anything to the power of zero is 1.

In this case, when t = 0, the expression (1/2)^0 equals to 1, so y equals the coefficient 10. Thus, the y-intercept is (0,10). This point is crucial for graphing because it gives a starting point from which the graph will decay exponentially. It's particularly significant because the y-intercept also reveals the initial quantity or value before any decay has started.

Domain and Range of Exponential Function

The domain and range are concepts that tell us what values the independent and dependent variables in a function can take, respectively. For exponential functions, the domain is pretty easy to remember – it's all real numbers because you can plug any real number into t and it'll work in the function.

However, the range is where things get interesting. Because exponential decay models approach but never reach zero, the range represents all positive values for y. Specifically, for our function y=10(1/2)^t, the range is y > 0. This tells you that the graph will never touch or cross the x-axis, which represents y = 0. Understanding the domain and range helps when graphing because you know to draw the curve starting at the y-intercept and nearing the x-axis without ever touching it, emphasizing the nature of exponential decay.