Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Find an exponential function of the form \(f(x)=b a^{x}\) that has the given \(y\)-intercept and passes through the point \(P\). \(y\)-intercept \(8 ; \quad P(3,1)\)

Step-by-Step Solution

Answer

The function is \( f(x) = 8 \left(\frac{1}{2}\right)^x \).

Step 1: Identify the y-intercept

The form of the exponential function is given by \( f(x) = b \cdot a^x \). The \( y \)-intercept occurs at \( x = 0 \). At this point, the function value \( f(0) = b \cdot a^0 = b \). We are given that the \( y \)-intercept is 8, so \( b = 8 \).

Step 2: Use the point P to find 'a'

The function passes through the point \( P(3, 1) \). Substitute \( x = 3 \) and \( f(x) = 1 \) into the function \( f(x) = 8 \cdot a^x \):\[ 8 \cdot a^3 = 1 \]Solve this equation for \( a \).

Step 3: Solve for the base 'a'

To solve \( 8 \cdot a^3 = 1 \), first divide both sides by 8:\[ a^3 = \frac{1}{8} \]Take the cube root of both sides to isolate \( a \):\[ a = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \].

Step 4: Write the exponential function

Substitute back the values of \( b = 8 \) and \( a = \frac{1}{2} \) into the exponential function form:\[ f(x) = 8 \left(\frac{1}{2}\right)^x \].This is the required exponential function.

Key Concepts

y-intercept

In the world of algebra and calculus, a "y-intercept" is a crucial concept for understanding graphs. It is the point where the graph of a function or equation crosses the y-axis. For any function, the y-axis is where the value of x is zero. Thus, when you substitute 0 for x in a function, the resulting output is the y-intercept.
In the context of exponential functions of the form \( f(x) = b \cdot a^x \), the y-intercept is found by evaluating \( f(0) \). Since anything raised to the power of 0 is 1, this simplifies to \( f(0) = b \cdot a^0 = b \). As identified in our problem, the given y-intercept is 8. This means that when \( x = 0 \), \( f(x) = 8 \), therefore \( b = 8 \).
This element of the function gives us the starting value or initial amount of "exponentially growing or decaying" quantities.

point on the graph

Another important aspect of understanding functions is knowing how they pass through specific points on the graph. Given a point, such as \( P(3,1) \), we understand it as a specific location the function must touch. It gives us the x-coordinate and the corresponding function value at that x.
Substituting this point into our exponential function helps us solve for the unknown desired value of the base 'a'. With the x-value of 3 and y-value (or function output) of 1, we insert these into the function equation: \( 1 = 8 \cdot a^3 \). Each point gives us valuable insight into the structure and behavior of the function.

solving equations

Solving equations is the bread and butter of mathematics. In this particular problem, we have an equation derived from the function and the given point on the graph: \( 8 \cdot a^3 = 1 \).
The first step in solving this equation is isolating the variable \( a \). This requires dividing both sides by 8, resulting in \( a^3 = \frac{1}{8} \).
Next, to solve for \( a \), we take the cube root of both sides, simplifying to \( a = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \). Solving exponential equations usually involves breaking down the equation step by step, ensuring variables are carefully isolated and calculated.

function form

An exponential function is a mathematical expression commonly appearing in real-life scenarios such as population growth and radioactive decay. It is generally expressed as \( f(x) = b \cdot a^x \).
In this form:

• \( b \) represents the y-intercept.
• \( a \) is the base, indicating the factor by which the function's output changes as x increases.

In our solution, we have identified \( b = 8 \) and \( a = \frac{1}{2} \). Thus, the complete function is \( f(x) = 8 \left( \frac{1}{2} \right)^x \).
This function form succinctly describes the rate of exponential change, showcasing how output diminishes or grows as the value of x is altered.