Transformations of functions: FAQ

When we shift a function horizontally, we are moving the entire graph of the function left or right. This is done by adding or subtracting a constant from the function's input. For example, to shift the function $f(x) = x^2$f, left parenthesis, x, right parenthesis, equals, x, squared three units to the left, we would write $f(x+3) = (x+3)^2$f, left parenthesis, x, plus, 3, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, squared.

Vertical shifting is similar to horizontal shifting, except we are moving the entire graph of the function up or down. This is done by adding or subtracting a constant from the function's output. For example, to shift the function $f(x) = x^2$f, left parenthesis, x, right parenthesis, equals, x, squared four units up, we would write $f(x) = x^2+4$f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 4.

When we reflect a function, we're flipping it over a specific line. For example, if we reflect a function over the $y$y-axis, we're flipping it from left to right. If we reflect a function over the $x$x-axis, we're flipping it from top to bottom.

When we scale a function, we're changing its size on the graph. For example, if we multiply a function by $2$2, we're making it twice as tall. If we multiply a function by $\dfrac{1}{2}$start fraction, 1, divided by, 2, end fraction, we're making it half as tall.

Yes! We use transformations in a variety of fields, like engineering, physics, and economics. For example, in physics, we often use transformations to change the units of a function in order to make it easier to work with. In economics, we might use transformations to help us compare different data sets.