Rope Drop 12/3

A vertical translation occurs when a function is shifted up or down on the graph. This is represented by adding or subtracting a constant from the function, such as $$f(x) + k$$ , where $$k$$ is a positive or negative number.

A horizontal translation occurs when a function is shifted left or right on the graph. This is represented by adding or subtracting a constant from the input variable, such as $$f(x + k)$$ , where $$k$$ is a positive or negative number.

A vertical dilation occurs when the output values of a function are stretched or compressed. This is represented by multiplying the function by a constant, such as $$k f(x)$$ , where $$k > 1$$ stretches the graph and $$0 < k < 1$$ compresses it.

A horizontal dilation occurs when the input values of a function are stretched or compressed. This is represented by changing the input variable, such as $$f(kx)$$ , where $$k > 1$$ compresses the graph and $$0 < k < 1$$ stretches it.

The transformation $$f(-x)$$ represents a reflection of the graph of the function across the y-axis.

The transformation $$-f(x)$$ represents a reflection of the graph of the function across the x-axis.

Adding a constant to the function, such as in $$f(x) + 3$$ , results in a vertical translation of the graph upwards by 3 units.

Subtracting a constant from the function, such as in $$f(x) - 2$$ , results in a vertical translation of the graph downwards by 2 units.

The transformation $$f(x + 2)$$ represents a horizontal translation of the graph to the left by 2 units.

The transformation $$f(x - 1)$$ represents a horizontal translation of the graph to the right by 1 unit.