Apply vertical and horizontal shifts, reflections and stretches to the graph of a function, and read a transformed function's equation from its parent (Functions).

What this topic is asking

A transformation moves or reshapes the graph of a "parent" function. The ACT tests how shifts, reflections and stretches change both the graph and the equation. The crucial rule is that changes inside the function (to $x$) act horizontally and counterintuitively, while changes outside act vertically and as expected.

Vertical and horizontal shifts

Shifts are the most common transformation.

The inside-shift rule trips many students: $f(x - 2)$ shifts right 2, even though it looks like a subtraction.

Reflections

A reflection flips the graph across an axis.

$-f(x)$ flips vertically (across the $x$-axis); $f(-x)$ flips horizontally (across the $y$-axis).

Stretches and compressions

Multiplying the function by a constant $a$ scales it vertically. If $a > 1$, the graph stretches away from the $x$-axis (taller); if $0 < a < 1$, it compresses toward the axis (flatter). So $2f(x)$ doubles every output, and $\frac{1}{2}f(x)$ halves them. A negative $a$ combines a stretch or compression with a reflection across the $x$-axis. On the ACT, vertical stretches are far more common than horizontal ones.

Combining transformations

Many questions apply two or more changes at once, as in $g(x) = -2f(x - 1) + 3$. Read it piece by piece: $f(x - 1)$ shifts right 1, the $-2$ reflects across the $x$-axis and stretches by 2, and $+3$ shifts up 3. Working from the inside out, and separating inside changes (horizontal) from outside changes (vertical), lets you describe any combination without confusion. Matching a described transformation to the right equation, or vice versa, is the typical ACT task.

How transformations move key points

A reliable way to apply a transformation is to track a single point. The vertex of $y = x^{2}$ sits at $(0, 0)$; under $g(x) = (x - 2)^{2} + 1$ it moves to $(2, 1)$, right 2 and up 1. Following the most recognisable point, a vertex, an intercept, an endpoint, through each step shows exactly where the new graph sits, without re-plotting the whole curve. This also catches sign errors: if your "right shift" sends the vertex left, you have misread an inside change.

Transformations of other parent functions

The same rules apply to every parent, not just the parabola. For the absolute-value parent $|x|$, the line $y = x$, the square-root $\sqrt{x}$, or an exponential $2^{x}$, adding outside shifts vertically, subtracting inside shifts right, a leading negative reflects across the $x$-axis, and a coefficient stretches vertically. Because the rules are identical across parents, learning them once on $x^{2}$ lets you transform any function the ACT presents, which is why the inside-versus-outside distinction is worth memorising rather than re-deriving each time.

Why the inside-outside rule matters

The reliable key to every transformation question is the distinction between inside and outside the function. Outside changes (adding, multiplying the whole function, negating it) behave as written and act vertically. Inside changes (to the $x$ before the function acts) behave oppositely in the horizontal direction. Holding that one rule firmly prevents the most common transformation error, mixing up horizontal and vertical shifts.

Try this

Q1. The graph of $g(x) = f(x) - 5$ is $f$ shifted how? [1 point]

• Cue. Down 5 (subtracting outside lowers every output).

Q2. If $f(x) = x^{2}$, write $f$ shifted left 3. [1 point]

• Cue. Add inside: $(x + 3)^{2}$ (left shift).