Problem 8 - Algebra 2

Question

Write an exponential function for the graph that passes through the given points. $$ (0,3) \text { and }(-1,6) $$

Step-by-Step Solution

Answer

The exponential function is \( y = 3 \left(\frac{1}{2}\right)^x \).

Step 1: Understanding Exponential Functions

An exponential function has the form \( y = ab^x \), where \( a \) is a constant, \( b \) is the base of the exponential function, and \( x \) is the independent variable. We need to determine the values of \( a \) and \( b \) using the given points.

Step 2: Use the First Point (0,3)

Substitute the first point \((0, 3)\) into the exponential function \( y = ab^x \):\[ 3 = ab^0 \]Since \( b^0 = 1 \), this simplifies to \( a = 3 \).Thus, the equation becomes \( y = 3b^x \).

Step 3: Use the Second Point (-1,6)

Substitute the second point \((-1, 6)\) into the equation \( y = 3b^x \):\[ 6 = 3b^{-1} \]Solving for \( b \), we multiply both sides by \( b \) to get \( 6b = 3 \).

Step 4: Solve for \( b \)

Rearrange the equation from Step 3:\[ 6b = 3 \]Divide both sides by 6:\[ b = \frac{3}{6} = \frac{1}{2} \].

Step 5: Write the Final Exponential Function

With \( a = 3 \) and \( b = \frac{1}{2} \), substitute these values back into the general form \( y = ab^x \) to get the final equation:\[ y = 3 \left(\frac{1}{2}\right)^x \].

Key Concepts

Point Slope Form

Point slope form is a linear equation representation technique frequently used in geometry and algebra. While it directly deals with linear equations, understanding it helps grasp more complex concepts like exponential functions. Linear equations in point slope form are written as:\[ y - y_1 = m(x - x_1) \]Here:

• \( y_1 \) and \( x_1 \) are the coordinates of a specific point on the line.
• \( m \) is the slope of the line, reflecting its steepness and direction.

In exponential contexts, unlike linear formulas, the rate of change isn't constant throughout. Once you understand how linear changes work through point slope, realizing how exponential functions exponentially "steep" gets easier. Practicing with both exponential and linear examples builds a robust foundation for understanding derivative relationships in calculus too.

Exponential Equation

Exponential equations are mathematical expressions where the variable appears in the exponent. This type of equation takes the form:\[ y = ab^x \]In our problem, the exponential equation was identified by using two points:

• (0, 3)
• (-1, 6)

Here's how we decode this pattern:

• The value at \( x = 0 \), which was 3, gives us \( a \) directly. That’s because any number raised to the zero power equals 1.
• The next task is using additional points to determine \( b \), which affects the growth or decay of the function.
• From our second point (-1, 6), careful substitution and solving revealed \( b = \frac{1}{2} \).

Understanding this framework helps solve complex problems by breaking them into solvable steps, using known points to find unknown values, as demonstrated in our solution step.

Graphing Exponential Functions

Graphing exponential functions is an intriguing process that visually represents how quickly values escalate or diminish over intervals. When plotting an exponential function, you will notice distinct characteristics:

• The graph passes through the point where \( x = 0 \) and \( y = a \), which in our problem was the point (0, 3).
• The growth or decay factor, \( b \), defines the graph’s trend: growth if \( b > 1 \), decay if \( 0 < b < 1 \).
• When \( b = \frac{1}{2} \), as found in our example, the curve gradually decreases—illustrating a decay process.

An exponential graph is usually continuously increasing or decreasing, never making sharp turns or leveling off completely. Recognizing these patterns assists in sketching or calculating function behavior over different time periods or input scales. As you practice, recognizing or anticipating the graphical representation of these equations becomes more intuitive.