Problem 555 - Precalculus

Question

Find an exponential equation that passes through the points \((0,4)\) and \((2,9)\).

Step-by-Step Solution

Answer

The equation is \(y = 4\left(\frac{3}{2}\right)^x\).

Step 1: Understanding the Exponential Equation

An exponential equation typically has the form \(y = ab^x\), where \(a\) is the initial value when \(x = 0\), and \(b\) is the base that determines the rate of growth or decay.

Step 2: Using the First Point to Find \(a\)

Since the point \((0, 4)\) lies on the curve, substituting \(x = 0\) and \(y = 4\) into \(y = ab^x\) gives us \(4 = ab^0\). Since anything raised to the power of zero equals 1, we get \(4 = a \times 1\) or simply \(a = 4\).

Step 3: Substituting \(a\) to Find \(b\) Using the Second Point

Now that we know \(a = 4\), substitute the point \((2, 9)\) into the equation \(y = 4b^x\). Thus, we have: \(9 = 4b^2\). We can solve for \(b^2\) by dividing both sides by 4, giving: \(b^2 = \frac{9}{4}\).

Step 4: Solving for \(b\)

To find \(b\), take the square root of both sides: \(b = \sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}\).

Step 5: Writing the Final Exponential Equation

Now that we have both values, \(a = 4\) and \(b = \frac{3}{2}\), substitute into the general exponential form: \(y = 4\left(\frac{3}{2}\right)^x\).

Key Concepts

Understanding Exponential Growth

Exponential growth is a fascinating and vital concept in mathematics, describing how quantities increase rapidly over time. When a quantity grows exponentially, its rate of growth is proportional to its current size. This means that as the amount increases, it grows faster and faster.
This can be observed in real-world situations such as population growth, the spread of diseases, and the amplification of investments.In mathematical terms, exponential growth can be represented by the formula \( y = ab^x \). Here, \( y \) signifies the current quantity, \( a \) denotes the initial value, \( b \) is the growth factor, and \( x \) is time or the exponent indicating how long the growth has been occurring.

• The variable \( a \) provides the starting point or the initial size of the quantity when \( x = 0 \).
• \( b \) determines how quickly the growth occurs; if \( b > 1 \), this indicates exponential growth.
• The exponent \( x \) reflects the duration or extent of growth. As \( x \) increases, the quantity \( y \) rises more swiftly if \( b > 1 \).

Exponential growth should not be confused with linear growth, where a fixed amount is added at regular intervals. In exponential growth, however, the addition pattern is proportional to the current size, visibly illustrating the power of accrued expansion.

The Role of the Initial Value in Exponential Functions

The initial value, often denoted as \( a \) in the exponential function \( y = ab^x \), plays a crucial role in establishing the starting point of the function. It's simply the value of \( y \) when \( x = 0 \), setting the base level from which growth or decay is measured. In an exponential model, the initial value acts as a multiplier affecting the entire scope of growth.
Understanding the initial value is key to unraveling the exponential equation.In the solution to the problem, we first used the point \((0, 4)\) to determine the initial value. Since any number raised to the power of zero is 1, the expression becomes \( y = a \times 1 \), leading directly to \( a = y \). With \( a \) known:

• The initial value (in this case, 4) represents the starting quantity before any changes begin to measure over time.
• It sets the benchmark from which exponential growth is calculated, making it critical in shaping the function's graph.

Recognizing the initial value helps in constructing and predicting future outcomes based on exponential behavior in various scenarios. It reflects the foundation upon which exponential changes build, scaling in proportion as time progresses.

Determining the Rate of Growth

In exponential functions, the rate of growth is determined by the base, denoted as \( b \) in the equation \( y = ab^x \). It represents how rapidly or slowly the quantity increases, setting the pace for the function's rise as the exponent \( x \) increases. The value of \( b \) is crucial, as it directly influences the steepness or explosiveness of the growth curve.
Let's dive deeper into understanding this rate.For the given exercise, the point \((2, 9)\) was used to find \( b \), ensuring both points satisfied the equation. Here's how it works:

• If \( b > 1 \), it indicates a true exponential growth; the quantity continually doubles or triples, depending on the value of \( b \).
• If \( 0 < b < 1 \), it implies exponential decay, where the quantity decreases over time.

To find \( b \), we solve the equation using another known point, leading to a calculated value of \( b = \frac{3}{2} \). This value of \( b \) reflects the growth rate, showing how quickly the quantity \( y \) boosts as \( x \) moves beyond zero.Understanding \( b \) helps in predicting how a system evolves, whether it's population, investments, or even biological phenomena, as it directly relates to the function's growth velocity and behavior.