Transformations of Functions

Front

What is a transformation of a function?

Back

A transformation of a function refers to the changes made to the graph of a function, which can include shifts, reflections, stretches, and compressions.

Front

What does it mean to shift a graph left or right?

Back

Shifting a graph left or right involves adding or subtracting a value from the input (x) of the function. For example, g(x) = f(x + c) shifts the graph left by c units, while g(x) = f(x - c) shifts it right by c units.

Front

What does it mean to reflect a graph over the x-axis?

Back

Reflecting a graph over the x-axis means that for every point (x, y) on the graph, the corresponding point on the reflected graph will be (x, -y). This is achieved by multiplying the function by -1.

Front

What does it mean to reflect a graph over the y-axis?

Back

Reflecting a graph over the y-axis means that for every point (x, y) on the graph, the corresponding point on the reflected graph will be (-x, y). This is achieved by replacing x with -x in the function.

Front

How does the equation f(x) = (x - h)^2 + k transform the graph of f(x) = x^2?

Back

The graph of f(x) = (x - h)^2 + k is shifted right by h units and up by k units from the graph of f(x) = x^2.

Front

What is the effect of a negative sign in front of a function, such as g(x) = -f(x)?

Back

A negative sign in front of a function reflects the graph over the x-axis.

Front

What does the term 'vertical shift' refer to in function transformations?

Back

A vertical shift refers to moving the graph of a function up or down. For example, g(x) = f(x) + k shifts the graph up by k units, while g(x) = f(x) - k shifts it down by k units.

Front

What is the transformation represented by g(x) = f(x + 3)?

Back

The transformation g(x) = f(x + 3) represents a horizontal shift of the graph of f(x) to the left by 3 units.

Front

What is the transformation represented by g(x) = f(x - 2)?

Back

The transformation g(x) = f(x - 2) represents a horizontal shift of the graph of f(x) to the right by 2 units.

Front

How do you describe the transformation from f(x) = x^2 to g(x) = x^2 - 4?

Back

The transformation from f(x) = x^2 to g(x) = x^2 - 4 is a vertical shift down by 4 units.