Exponential Functions - Graphs and Models

What is an exponential function?

• A mathematical function of the form f(x) = a + b^x, where a is a constant and b is the base.

• A mathematical function of the form f(x) = a * b^x, where a is a constant, b is the base (a positive real number), and x is the exponent.

• A function that grows linearly with respect to x.

• A function that has a constant rate of change.

What is the general form of an exponential function?

• f(x) = a * b^(x - h) + k

• f(x) = a * e^(bx) + c

• f(x) = a * x^b + c

• f(x) = a + b * log(x)

What does a vertical translation up by 2 units mean for the function f(x)?

• It means the new function can be expressed as f(x) + 2, shifting the entire graph of f(x) upward by 2 units.

• It means the new function can be expressed as f(x) - 2, shifting the entire graph of f(x) downward by 2 units.

• It means the new function can be expressed as 2f(x), stretching the graph vertically by a factor of 2.

• It means the new function can be expressed as f(x + 2), shifting the graph horizontally to the right by 2 units.

What does the y-intercept of an exponential function represent?

• The value of the function when x = 0

• The slope of the function at x = 0

• The maximum value of the function

• The minimum value of the function

What is the difference between exponential growth and exponential decay?

• Exponential growth occurs when the base b > 1, leading to an increase in the function's value as x increases.

• Exponential decay occurs when the base b > 1, leading to an increase in the function's value as x increases.

• Exponential growth and decay are the same, both leading to a decrease in the function's value as x increases.

• Exponential growth occurs when 0 < b < 1, leading to a decrease in the function's value as x increases.

How do transformations affect the graph of an exponential function?

• Transformations change the base of the exponential function.

• Transformations alter the position and shape of the graph without changing its fundamental exponential nature.

• Transformations only affect the y-intercept of the graph.

• Transformations make the graph linear instead of exponential.

What is the effect of a negative exponent in an exponential function?

• A negative exponent indicates a reciprocal function, meaning f(x) = a * b^(-x) can be rewritten as f(x) = a / b^x, which represents exponential decay.

• A negative exponent results in a constant function that does not change with x.

• A negative exponent indicates that the function will always increase regardless of the value of x.

• A negative exponent has no effect on the function and can be ignored.

What is a horizontal translation in the context of exponential functions?

• A horizontal translation involves shifting the graph of the function left or right by adding or subtracting a constant from the x-variable.

• A horizontal translation involves changing the base of the exponential function.

• A horizontal translation refers to the vertical movement of the graph along the y-axis.

• A horizontal translation is the reflection of the graph across the y-axis.

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

• The base determines the rate of growth or decay; a larger base results in faster growth, while a base between 0 and 1 results in decay.

• The base has no effect on the function's output.

• The base only affects the horizontal shift of the graph.

• The base determines the maximum value of the function.

How do you determine the y-intercept of the function f(x) = 2 * 3^x?

• f(0) = 2 * 3^0 = 2 * 1 = 2

• f(0) = 2 * 3^0 = 2 * 0 = 0

• f(0) = 2 * 3^0 = 2 * 3 = 6

• f(0) = 2 * 3^1 = 2 * 3 = 6