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Transformations: Shifts, Stretches, and Reflections

Take one parent graph and move it anywhere. Learn how adding, subtracting, and multiplying inside or outside a function slides, flips, and stretches its graph — and how to combine several moves in the right order.

Shifts: sliding the whole graph

Start from a parent function like f(x) = x^2, whose graph is a parabola with its vertex at the origin. A vertical shift adds a number *outside* the function: f(x) + k slides the graph up by k (down if k is negative). A horizontal shift adds *inside*: f(x − h) slides the graph right by h — note the backwards-feeling sign.

Parent: f(x) = x^2 (vertex at (0,0))

f(x) + 4   = x^2 + 4        -> up 4         vertex (0, 4)
f(x) - 4   = x^2 - 4        -> down 4       vertex (0, -4)
f(x - 3)   = (x - 3)^2      -> right 3      vertex (3, 0)
f(x + 3)   = (x + 3)^2      -> left 3       vertex (-3, 0)
f(x-3) + 4 = (x-3)^2 + 4    -> right 3, up 4   vertex (3, 4)
Outside the function moves it vertically; inside moves it horizontally, with the sign reversed.

Stretches and reflections

Multiplying *outside*, a·f(x), scales the graph vertically: a = 3 stretches it taller, a = 1/2 squashes it flatter. A negative multiplier gives a reflection: −f(x) flips the graph upside down across the x-axis, while f(−x) flips it left-to-right across the y-axis. (For an even function, that left-right flip changes nothing.)

Parent: f(x) = x^2

3 f(x)  = 3x^2      -> vertical stretch (3x taller, narrower)
(1/2)f(x)= (1/2)x^2  -> vertical shrink (flatter, wider)
-f(x)   = -x^2      -> reflect across x-axis (opens downward)
f(-x)   = (-x)^2 = x^2  -> reflect across y-axis: unchanged (even!)
A multiplier sets the steepness; a minus sign flips the graph over an axis.

Combining moves and reading vertex form

Real problems stack several moves. The safe order is: handle the *inside* (horizontal shift and reflection) first, then the *outside* stretch, then the *outside* shift. The quadratic vertex form g(x) = a(x − h)^2 + k packages all of this: it is the parent x^2 stretched by a, shifted right h, up k, with vertex at (h, k) and axis of symmetry x = h.

  1. Read g(x) = −2(x − 1)^2 + 5 against the template a(x − h)^2 + k: here a = −2, h = 1, k = 5.
  2. Start with x^2; shift right 1 (the inside x − 1).
  3. Stretch by 2 and reflect across the x-axis (the −2): the parabola opens downward and is steeper.
  4. Shift up 5 (the + 5). Final vertex: (1, 5), axis of symmetry x = 1, opening downward.