What is the exponential function with a concave up graph and two given points?

In summary, the conversation discusses finding the exponential function f(x) = Ca^x from a given graph. It is mentioned that the graph has a concave up line and intersects the y-axis at approximately 1/2. The point (1,6) on the graph is used to find a similar equation, and the point (3,24) is used to create two equations with two unknowns. It is then noted that the graph does not intersect the y-axis at 1/2, but rather at y=3/2. This information helps in solving the problem.
  • #1
JonF
621
1
I have no idea how to do this problem or where to even start: Find the exponential function f(x) = Ca^x who’s graph is given. The graph has a concave up line, that intersects the y-axis at what looks like ½ (no exact intersection is given) and has the two points (1,6) and (3,24).
 
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  • #2
That the point (1, 6) lies on the graph means that f(1) = 6. But f(x) = Ca^x, so f(1) = Ca^1 = Ca = 6. You can find a similar equation using the other point. You now have two equations and two unknowns. Can you take it from there?
 
  • #3
However, the graph of y= Cax, passing through (1,6) and (3,24) doesn't cross the y-axis anywhere near 1/2!
 
  • #4
HallsofIvy said:
However, the graph of y= Cax, passing through (1,6) and (3,24) doesn't cross the y-axis anywhere near 1/2!

It crosses at y=3/2.
 
  • #5
That helps a lot, thank you. Sorry about the ½ thing, didn’t do a very good job of eyeballing it.
 

What is an exponential function?

An exponential function is a mathematical function where the independent variable (usually denoted as x) is in the exponent. The general form of an exponential function is y = ab^x, where a is the initial value and b is the base. These functions are used to model phenomena that grow or decay at a constant rate.

What are some real-life examples of exponential functions?

Some common examples of exponential functions in real life include population growth, compound interest, radioactive decay, and bacterial growth. These phenomena can be modeled using exponential functions because they exhibit a constant rate of change.

What is the difference between exponential growth and exponential decay?

Exponential growth occurs when the base of an exponential function is greater than 1, resulting in the function increasing at an increasing rate. On the other hand, exponential decay occurs when the base is between 0 and 1, causing the function to decrease at a decreasing rate.

How do you graph an exponential function?

To graph an exponential function, you can create a table of values by choosing different values for x and plugging them into the function to find the corresponding y values. Then, plot the points on a graph and connect them with a smooth curve. Alternatively, you can use the properties of exponential functions, such as the asymptote and the y-intercept, to sketch the graph without needing to plot specific points.

What are some real-life applications of exponential functions in science?

Exponential functions are commonly used in various fields of science, such as biology, chemistry, physics, and finance. They can be used to model population growth, radioactive decay, chemical reactions, and the spread of diseases. In finance, exponential functions are used to calculate compound interest and to model stock market trends. They are also used in technology, such as in the growth of computer processing speed over time.

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