Question

Write an exponential function \(y=a b^x\) whose graph passes through the given points. \((3,27),(5,243)\)

Short answer

The exponential function that passes through the points (3,27) and (5,243) is \(y = 3^x\).

Step-by-step solution

Use the given points to set up the equations

Let's substitute the points (3,27) and (5,243) into the general form of the equation \(y = ab^x\). This will result in two equations:\[27 = ab^3\] and \[243 = ab^5\]. Although we have two unknowns, we can call the first equation (1) and the second one (2). They can be solved simultaneously for 'a' and 'b'.

Isolate 'a' and set the equations equal to each other

From equation (1), we can isolate 'a' by dividing both sides by \(b^3\) to get \(a=\frac{27}{b^3}\). And, for equation (2), we can isolate 'a' by dividing both sides by \(b^5\) to get \(a=\frac{243}{b^5}\). Now, we can set the two equations equal to each other which will give \(\frac{27}{b^3} = \frac{243}{b^5}\).

Solve for 'b'

To solve for 'b', we can simplify our equation from step 2 by multiplying each side by \(b^5\) and by \(b^3\), that then gives \(27b^2 = 243\). Solving for 'b' now by dividing through by 27 we find that \(b = 3\).

Find 'a' using the value of 'b'

We can substitute the value of 'b' into equation (1) to get \(a = 27 /3^3\), which simplifies to \(a = 1\).

Write the exponential function

Now that we have both 'a' and 'b', we can write the function as \(y = a*b^x = 1*3^x\), or simply \(y = 3^x\).

Key concepts

Graphing Solutions

When you are tasked with graphing an exponential function, you need to visualize the relationship between the variables. In this example, the exponential function is of the form \( y = ab^x \). Each graph of such an equation will display a unique curve, growing or decreasing depending on the value of the base \( b \).

The points given are (3, 27) and (5, 243). Plot these on the coordinate plane, and they should align with the curve of your function. Notice how as \(x\) increases, \(y\) increases rapidly, indicating exponential growth. This growth is a key characteristic of exponential functions:

• If \(0 < b < 1\), the graph will decay exponentially.
• If \(b > 1\), as in our case \(b = 3\), the function will grow exponentially.

Choosing the correct values for \(a\) and \(b\) ensures that your graph accurately represents the exponential function you are modeling.

Solving Equations

Solving equations in the context of exponential functions often involves working with exponents and logarithms. In the given solution, the exponential function is constructed using two given points, which is essential for determining the constants \(a\) and \(b\).

Here's the step-by-step breakdown:

• With points (3,27) and (5,243), substitute into \(y = ab^x\) resulting in two equations.
• Isolate \(a\) in both equations: \(a = \frac{27}{b^3}\) and \(a = \frac{243}{b^5}\).
• Set the expressions for \(a\) equal to solve for \(b\).

This leads to simplifying and finding distinct values for \(b\), further uncovering \(b = 3\). With \(b\) known, substitute back to find \(a\), completing the solution.

Algebraic Expressions

Algebraic expressions form the backbone of manipulating and solving exponential equations. With exponential functions, clear steps must be taken to rearrange and solve for missing elements, usually contained within an expression.

For the exercise at hand, the expression \( y = ab^x \) represents our function. In solving it, you will encounter other forms of expressions such as \(\frac{27}{b^3}\) and \(\frac{243}{b^5}\). It's important to:

• Recognize the power of exponentiation in rewriting expressions.
• Understand how isolating variables in expressions leads to simpler forms for solving.

By equating these two isolated forms, you can solve for the unknown variables, demonstrating the usefulness and power of algebra in creating a straightforward path to the solution.

Mathematical Modeling

Mathematical modeling with exponential functions involves forming an equation that represents real-world scenarios or hypothetical situations. Modelling provides a way to predict outcomes based on variable manipulations.

With the task of determining \(y = ab^x\) from points (3, 27) and (5, 243):

• The model \(y = 3^x\) reflects a situation where output values grow exponentially as a result of increasing input \(x\).
• Such exponential forms often model scenarios regarding population growth, compound interest, and natural phenomena fit.
  
  

Understanding this process gives insight into predicting and interpreting behaviors in diverse settings, emphasizing the importance of grasping how exponential functions apply outside of theoretical mathematics.