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Chapter 9: Problem 29

Write an exponential function for the graph that passes through the given points. $$ (0,3) \text { and }(1,15) $$

Short Answer

Expert verified
The function is \( y = 3 \times 5^x \).

Step by step solution

01

Understand the Exponential Function Form

An exponential function can be represented by the equation \( y = ab^x \), where \( a \) is the initial value and \( b \) is the base of the exponential function.
02

Substitute the First Point

The given point \( (0, 3) \) tells us that when \( x = 0 \), \( y = 3 \). Substitute these values into the equation: \( 3 = ab^0 \). Since \( b^0 = 1 \), we have \( a = 3 \). This gives us the equation \( y = 3b^x \).
03

Substitute the Second Point

The second point is \( (1, 15) \). Using \( y = 3b^x \), substitute \( x = 1 \) and \( y = 15 \). This gives: \( 15 = 3b^1 \) which simplifies to \( 15 = 3b \).
04

Solve for Base \( b \)

Divide both sides of the equation \( 15 = 3b \) by 3 to solve for \( b \). This results in \( b = 5 \).
05

Write the Exponential Function

Now that we have \( a = 3 \) and \( b = 5 \), substitute these back into the exponential function format to get the final equation: \( y = 3 \cdot 5^x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Exponential growth is a concept where a quantity increases by a consistent factor over equal increments of time. In simple terms, it means something grows faster and faster as it gets larger. For example, if you start with $100 and it grows by 5% every year, the amount it grows each year becomes larger. Key points about exponential growth include:
  • The rate of growth becomes quicker over time.
  • It often models processes like population growth, compound interest, and spread of diseases.
  • This growth is represented graphically as a curve that starts slowly and steepens rapidly.
If we take a function like the one given in our problem, where the graph passes through the points (0,3) and (1,15), it's essential to understand that this is a simple case of exponential growth. With each step, the value of y increases by multiplying the previous value by a base factor鈥攅mphasizing the principle of rapid growth.
Function Form
The standard form of an exponential function can be expressed as \[ y = ab^x \] where \( a \) represents the initial value, and \( b \) is the base of the exponential function, which dictates the rate of growth. Understanding this form is crucial to solving exponential problems. Here's why:
  • \( a \) (the initial value) is what you start with, like starting capital or initial population.
  • \( b \) (the base) is what you multiply by for each increment in \( x \), determining the growth rate.
For instance, in the exercise, the initial point (0,3) indicates that \( a = 3 \). Thus, the function starts with a value of 3 when \( x \) is zero. This function form enables us to efficiently describe how the values change and provide an easy way to manipulate growth scenarios.
Equation Solving
Solving an exponential equation involves using given points to find unknown parameters in the function form. For our task, we are given two points: (0,3) and (1,15). Steps to solve:
  • Start by substituting the initial point (for example, \(x = 0\)) into the function to find \(a\). Since the problem states when \(x = 0, y = 3\), our equation becomes \(3 = ab^0\). Since any number to the power of zero is one, it simplifies to \(a = 3\).
  • Next, use the second point to find \(b\), which is the growth factor. Substitute into \(y = 3b^x\) using point (1,15). Therefore, \(15 = 3b^1\), leading to \(15 = 3b\).
  • Solving for \(b\) requires dividing both sides by 3, yielding \(b = 5\).
By following these steps, we identify the key components of the exponential function, completing the equation \(y = 3 \, \cdot \, 5^x\), which models the growth behavior outlined by the points.
Graph Interpretation
Interpreting the graph of an exponential function involves understanding how changes in the equation's parameters affect its shape. Exponential graphs typically start with a relatively flat section, then rapidly increase in steepness.Main features of the exponential graph:
  • The y-intercept, where the function touches the y-axis, represents the initial value \(a\). In our example, \(y = 3 \cdot 5^x\), the y-intercept is 3.
  • The base \(b\) influences the curve's steepness. Larger values of \(b\) produce steeper graphs.
  • As \(x\) increases, \(y\) grows more quickly, showing the rapid expansion characteristic of exponential functions.
For the graph of the exponential function determined from the provided points, the initial point (0,3) indicates a y-intercept at 3. As \(x\) becomes 1, moving to point (1,15), the curve reflects the multiplicative growth by the base of 5, showcasing the function's steep and expanding nature.

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Most popular questions from this chapter

For Exercises 55 and 56 , use the following information. If you deposit \(P\) dollars into a bank account paying an annual interest rate \(r\) (expressed as a decimal), with \(n\) interest payments each year, the amount \(A\) you would have after \(t\) years is \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Marta places \(\$ 100\) in a savings account earning 2\(\%\) annual interest, compounded quarterly. How long will it take for Marta's money to double?

For Exercises \(36-38,\) use the graph at the right. U.S. growers were forecasted to produce 264 million pounds of pecans in 2003 . Write the number of pounds of pecans forecasted by U.S. growers in 2003 in scientific notation.

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 16 $$

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{50} 23\)

Write an equivalent exponential or logarithmic equation. $$ \ln 4+2 \ln x=8 $$
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