Problem 11 - Algebra 2

Question

Write an exponential function \(y=a b^{x}\) for a graph that includes the given points. $$ (2,122.5),(3,857.5) $$

Step-by-Step Solution

Answer

The exponential function that includes the given points is \(y = 2.5 \cdot 7^x\).

Step 1: Setting Up the Exponential Equations

First, plug in the x and y values from the two given points into the exponential function to derive two equations. For the point (2,122.5), the equation becomes \(122.5 = ab^2\). For the point (3,857.5), the equation becomes \(857.5 = ab^3\).

Step 2: Solving the Equations

Solve the first equation for 'a' to get \(a = 122.5 \div b^2\). Substitute this into the second equation to get \(857.5 = (122.5 \div b^2) \cdot b^3\). Simplify this equation by multiplying the right side to give \(857.5 = 122.5b\). Solving for 'b' gives \(b = 857.5 \div 122.5 \approx 7\).

Step 3: Solve for 'a'

Now substitute 'b' back into the first equation to solve for 'a'. This gives \(a = 122.5 \div (7^2) \approx 2.5\).

Step 4: Getting the Exponential Function

Substitute 'a' and 'b' into the exponential function to get \(y = 2.5 \cdot 7^x\).

Key Concepts

Equation Solving

Solving equations involves determining the unknown values that make the equation true. In this problem, we have two equations derived from the points on the graph of an exponential function. The key is to solve these simultaneously to find values for the variables \(a\) and \(b\).

Here’s a quick breakdown of the method used:

• Use the form of the exponential function \(y = ab^x\) and plug in the given points as coordinates to form two equations.
• Solve one of the equations for one variable (like \(a\)) and substitute it into the other equation. This reduces the number of variables.
• Continue the process using algebraic manipulation to find the values of both variables.

Equation solving can require steps like rewriting and rearranging the equations, which is an essential skill in solving any math problem efficiently.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It’s the foundation for equation solving and further mathematical concepts like functions.

In this exercise, algebra helps us:

• Rearrange equations: Algebra allows us to move terms from one side of an equation to another to isolate variables, aiding us in finding solutions.
• Perform operations like division and multiplication to simplify complex expressions.
• Apply logarithmic identities where needed in more advanced problems, though not necessary here.

Understanding algebra lays the groundwork for developing problem-solving skills across various math topics. In exponential equations, recognizing patterns and implementing algebraic principles leads to the correct solutions.

Function Graphing

Graphing a function means visually representing the relationship between input values \(x\) and their corresponding output values \(y = ab^x\). In exponential functions like the one in this exercise, graphing helps visualize how rapidly the function grows depending on the values of \(a\) and \(b\).

Some basics of graphing exponential functions include:

• The base \(b\) determines the growth rate: If \(b > 1\), the function shows exponential growth, and if \(b < 1\), it shows exponential decay.
• The constant \(a\) affects the vertical stretch or compression and the y-intercept of the graph.
• Exponential graphs are not symmetrical, which is different from other graph types like parabolas.

By understanding graphing, students can easily predict how changes in \(a\) and \(b\) will affect the graph's appearance, helping them solve and verify equations effectively.