Transformations of Exponential Functions

Front

What is a vertical shift in the context of exponential functions?

Back

A vertical shift occurs when the entire graph of the function moves up or down on the y-axis. For example, g(x) = f(x) - k shifts the graph down by k units.

Front

How does a horizontal shift affect the graph of an exponential function?

Back

A horizontal shift moves the graph left or right on the x-axis. For example, g(x) = f(x - h) shifts the graph right by h units, while g(x) = f(x + h) shifts it left by h units.

Front

What does it mean for an exponential function to reflect over the x-axis?

Back

Reflecting over the x-axis means that the output values of the function are negated. For example, if f(x) = 2^x, then g(x) = -f(x) = -2^x reflects the graph over the x-axis.

Front

What is the parent function of exponential functions?

Back

The parent function of exponential functions is f(x) = a^x, where a > 0. This function has a characteristic shape that increases or decreases rapidly.

Front

What transformation occurs when you add a constant to an exponential function?

Back

Adding a constant to an exponential function results in a vertical shift. For example, g(x) = f(x) + k shifts the graph up by k units.

Front

What does it mean to translate a function 5 units left?

Back

Translating a function 5 units left means to replace x with (x + 5) in the function's equation. For example, if f(x) = 2^x, then g(x) = 2^(x + 5) represents the translation.

Front

How do you determine the transformations applied to the function g(x) = -2^(x + 3) - 6?

Back

The transformations include a reflection over the x-axis (due to the negative sign), a horizontal shift left 3 units (due to (x + 3)), and a vertical shift down 6 units.

Front

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

Back

A negative coefficient reflects the graph of the function over the x-axis. For example, g(x) = -f(x) will invert the output values of f(x).

Front

How do you write a function that is translated 4 units up?

Back

To translate a function 4 units up, you add 4 to the function's output. For example, if f(x) = 2^x, then g(x) = 2^x + 4 represents the translation.

Front

What is a horizontal translation right 1 unit?

Back

A horizontal translation right 1 unit means to replace x with (x - 1) in the function's equation. For example, if f(x) = 2^x, then g(x) = 2^(x - 1) represents the translation.