Identify the effect on the graph of a function of replacing f(x) with f(x) + k, f(x - h), and a times f(x), including vertical and horizontal translations, stretches, compressions, and reflections (MA.912.F.2.1, MA.912.F.2.2).

What this topic is asking

MA.912.F.2 asks how changing a function's formula changes its graph. You learn to read three moves: adding outside the function ($f(x) + k$) shifts it vertically, changing the input ($f(x - h)$) shifts it horizontally, and multiplying ($a \cdot f(x)$) stretches, compresses, or reflects it. The B.E.S.T. Algebra 1 EOC tests these on parabolas, lines, exponentials, and absolute-value graphs.

Vertical shifts (outside the function)

Adding a constant after the function moves the whole graph up or down:

If $k > 0$ the graph rises by $k$; if $k < 0$ it drops by $|k|$. This is intuitive: every output gains $k$. So $f(x) = x^2$ becomes $f(x) + 3 = x^2 + 3$, the same parabola lifted 3 units.

Horizontal shifts (inside the function)

Changing the input moves the graph left or right, and the direction is opposite to the sign you see:

So $(x - 2)^2$ moves the parabola right 2, and $(x + 5)^2$ moves it left 5. This reversal is the single most error-prone idea in the topic.

Stretches, compressions, and reflections

Multiplying the function by a constant scales it vertically:

• $|a| > 1$: vertical stretch (taller, steeper).
• $0 < |a| < 1$: vertical compression (shorter, flatter).
• $a < 0$: reflection across the $x$-axis (flips upside down), in addition to any stretch.

How the B.E.S.T. EOC examines this topic

• Multiple choice and editing task. Describe the transformation from $f$ to $g$, or pick the equation matching a described shift.
• GRID and matching. Match transformed equations to graphs, or plot the new vertex.
• Multiselect. Select all transformations applied to a parent function.

A clarifying idea: outside changes affect the output (the $y$-direction) and behave intuitively, while inside changes affect the input (the $x$-direction) and behave in reverse. Sorting each constant into "inside" or "outside" first tells you which rule and which direction to use.

Why horizontal shifts go the opposite way

The reversal is not arbitrary; it follows from when each part of the graph appears. Take $g(x) = f(x - 2)$. To get the output that $f$ produced at input $0$, you now need $x - 2 = 0$, that is $x = 2$. So the feature that sat at $x = 0$ on $f$ now sits at $x = 2$ on $g$, a shift to the right, even though you see a minus sign. The input must be made larger to "cancel" the subtraction, which pushes every feature in the positive direction. The same logic shows $f(x + 2)$ shifts left, because you need $x = -2$ to recover the original input of $0$. Understanding this lets you reconstruct the rule instead of memorizing a direction that feels backward.

Connecting to vertex form

This topic is the engine behind the vertex form of a quadratic, $f(x) = a(x - h)^2 + k$. Reading it as transformations of $y = x^2$: $h$ is the horizontal shift (right for positive $h$), $k$ is the vertical shift, and $a$ is the stretch and possible reflection. That is exactly why the vertex sits at $(h, k)$, the point $(0, 0)$ of $x^2$ moved by $h$ and $k$. Seeing vertex form as transformations means you can graph any quadratic from its form without plotting points, which is valuable when the on-screen calculator cannot graph for you.

Try this

Q1. How is $g(x) = f(x) + 7$ related to $f$? [1 point]

• Cue. Shifted up 7 units.

Q2. Starting from $y = x^2$, give the vertex of $y = (x - 4)^2 - 1$. [1 point]

• Cue. Right 4, down 1, so the vertex is $(4, -1)$.