Problem 65 - Algebra 2

Question

Write an exponential equation \(y=a b^{x}\) whose graph passes through the given points. \(\left(0, \frac{1}{2}\right)\) and \((2,8)\)

Step-by-Step Solution

Answer

The exponential equation which passes through the points (0, 1/2) and (2,8) is \( y = 2^{x+1} \)

Step 1: Identify 'a'

The 'a' value in our exponential equation is the y-value when x = 0. Looking at our points, we get a = 1/2 when x = 0 from the point (0, 1/2). So a = 1/2.

Step 2: Substitute point (2,8) and 'a' value into the equation

We now know the 'a' value. The second point (2,8) can be used to find the 'b' value by substituting these values into our equation. Therefore, our equation becomes: 8 = 1/2 * b^2

Step 3: Solve for 'b'

Solve the equation to find 'b': 8 = 1/2 * b^2. Multiply both sides of the equation by 2 to get: 16 = b^2. Taking square root on both sides, we find b = 4.

Step 4: Formulate the exponential equation

Now, we can plug 'a' and 'b' into our equation to get the final exponential formula which is y = (1/2)*4^{x}, simplifying this we get: y = 2^{x+1}

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. The general form is given by \( y = a b^x \), where \( a \) and \( b \) are constants, and \( x \) is the variable exponent.

• The base \( b \) is a positive real number, crucial for shaping the curve of the graph.
• The constant \( a \) acts as the initial value or the y-intercept when \( x = 0 \).

An interesting property of exponential functions is that they can model growth or decay processes, like population growth or radioactive decay.When \( b > 1 \), the function represents exponential growth, having a rapid increase. Conversely, when \( 0 < b < 1 \), the function depicts exponential decay, showing a rapid decrease. Understanding these features helps predict and interpret various real-world scenarios modeled by these functions.

Solving Equations

Solving equations involves finding the value of unknown elements that satisfy the equation. In the case of exponential equations like \( y = a b^x \), you often need to determine the values of \( a \) and \( b \) given some data points. Consider solving this type of equation:

• Firstly, identify the constant \( a \) by using the y-value at \( x = 0 \).
• Use another point to determine the base \( b \) by substituting \( a \) into the equation and solving for \( b \).

For instance, from the point \((0, \frac{1}{2}) \), \( a \) is identified as \( \frac{1}{2} \). Using another point, say \((2,8)\), the equation becomes \( 8 = \frac{1}{2} b^2 \). Solving this gives us \( b = 4 \).Finally, formulate the exponential equation by placing determined values back into the formula, yielding a complete function to describe the relationship between variables.

Coordinate Points

Coordinates in mathematics refer to numerical values that specify positions on a graph. In the context of exponential functions, given points are crucial for determining the unknown parameters of the equation. Each coordinate is an ordered pair \((x, y)\) that provides specific information:

• The \( x \)-coordinate indicates the position along the horizontal axis.
• The \( y \)-coordinate shows the height or position along the vertical axis.

For example, consider points \((0, \frac{1}{2})\) and \((2,8)\):- The first point \((0, \frac{1}{2})\) immediately tells us the value of \( a \), since it is the y-intercept of the exponential graph.- The second point \((2,8)\) helps in solving for \( b \), crucial for defining the slope and direction of the exponential curve.These coordinate points are indispensable as they provide the essential data needed to construct the exponential function.