Writing Exponential Equations from Two Points

Front

What is the general form of an exponential function?

Back

The general form of an exponential function is y = ab^x, where a is a constant, b is the base, and x is the exponent.

Front

How do you determine the value of 'a' in the exponential function y = ab^x when given a point (x, y)?

Back

To determine 'a', substitute the x and y values from the point into the equation and solve for 'a'.

Front

What does the point (0, a) represent in an exponential function y = ab^x?

Back

The point (0, a) represents the y-intercept of the exponential function, which is the value of the function when x = 0.

Front

How can you find the base 'b' in the exponential function y = ab^x using two points?

Back

To find 'b', use the formula b = (y2/y1)^(1/(x2-x1)) where (x1, y1) and (x2, y2) are the two points.

Front

Given points (0, 6) and (2, 96), how do you find the exponential function?

Back

First, find 'a' using (0, 6) to get a = 6. Then, use (2, 96) to find 'b' by solving 96 = 6b^2, leading to b = 4. The function is y = 6(4^x).

Front

What is the significance of the base 'b' in an exponential function?

Back

The base 'b' determines the growth or decay rate of the function. If b > 1, the function grows; if 0 < b < 1, the function decays.

Front

How do you write an exponential function from two points?

Back

1. Identify 'a' from the first point. 2. Use the second point to create an equation to solve for 'b'. 3. Write the function in the form y = ab^x.

Front

What is the relationship between the points (0, a) and (x, y) in an exponential function?

Back

The point (0, a) gives the initial value, while (x, y) shows how the function behaves at other values of x based on the growth/decay rate.

Front

If an exponential function passes through (0, 4) and (2, 324), what is the function?

Back

The function is y = 4(9^x), where a = 4 and b = 9.

Front

What is the formula to find the exponential function given two points (x1, y1) and (x2, y2)?

Back

1. Find 'a' using y1 = ab^x1. 2. Use y2 = ab^x2 to create a second equation. 3. Solve the system of equations to find 'a' and 'b'.