Problem 23 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(y=(0.95)^x\)

Step-by-Step Solution

Answer

The given exercise represents an exponential function with a base of 0.95. The value of y decreases as x increases.

Step 1: Identify the form of equation

The given equation \(y=(0.95)^x\) represents an exponential function profile where the base 'a' is 0.95 and x is the variable.

Step 2: Evaluate exponential function

Calculate \(y\) for the variable \(x\). For any given value of x, y can be calculated. In this case let's suppose x=1, thus y can be calculated as \(y= (0.95)^1 = 0.95\)

Step 3: Additional Values

To fully understand the curve, additional values should be evaluated. For x=0, since any non-zero number raised to the power 0 is 1, \(y = (0.95)^0 = 1\). For x=2, \(y = (0.95)^2 = 0.9025\). Thus this gives an idea of how the curve of the function decreases as \(x\) increases.

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In mathematical terms, this is represented using an exponential function, which is a function of the form \( y = a^x \), where \( 0 < a < 1 \).
For the given function \( y = (0.95)^x \), the base \( 0.95 \) signifies that each unit increase in \( x \) results in the function's output reducing to 95% of its previous value. This means the function is experiencing decay, as the value of \( y \) decreases with increasing \( x \).
Some common real-world examples include depreciation of a car's value, the cooling of a cup of coffee, or radioactive decay. In every scenario, the rate at which the decay occurs mirrors the base of the exponential function: the closer \( a \) is to zero, the faster the decline.

Function Evaluation

Function evaluation involves calculating the output of a function for given inputs. For exponential functions like \( y = (0.95)^x \), it helps determine how the function behaves at several points.
To evaluate an exponential function, substitute the value of \( x \) into the function and solve for \( y \). For example:

• When \( x = 0 \), \( y = (0.95)^0 = 1 \). This result makes sense because any non-zero number raised to the power of 0 is 1.
• When \( x = 1 \), \( y = (0.95)^1 = 0.95 \). At this point, the function has decayed by 5%.
• When \( x = 2 \), \( y = (0.95)^2 = 0.9025 \). Here, the function has continued to decay, showcasing how values decrease further with increasing \( x \).

Evaluating functions at multiple points provides insight into their overall pattern, making it easier to predict future values.

Graphing Exponential Functions

Graphing exponential functions helps visualize their behavior over a range of inputs. The graph of \( y = (0.95)^x \) is a declining curve which starts at \( y = 1 \) when \( x = 0 \) and approaches but never quite reaches zero as \( x \) becomes very large.
An exponential decay graph features several distinct characteristics:

• It is always above the \( x \)-axis since \( a^x > 0 \) for all real numbers \( a \) with \( a > 0 \).
• The curve approaches the \( x \)-axis asymptotically, meaning it gets infinitely close but never touches it.
• It shifts downwards, getting flatter as \( x \) increases, indicating a slow down in the rate of decay.

To effectively graph it, begin by plotting calculated points such as (0, 1), (1, 0.95), and (2, 0.9025). Connecting these points with a smooth, continuous curve will display the function's decay characteristics and assist in understanding its behavior better.