Packets  /  Function Transformations
Grades 9-11Algebra

Function Transformations

Master how graphs move: vertical and horizontal shifts, reflections, and stretches of parent functions. Learn each transformation, practice describing and writing transformed equations with instant feedback, and beat the 60-second challenge.

1Vertical Shifts

Key idea: Adding a number OUTSIDE the function, g(x) = f(x) + k, moves the graph vertically — up if k is positive, down if k is negative.
  1. Find the number added or subtracted outside, after f(x) is computed.
  2. If it is positive, the graph shifts UP by that many units.
  3. If it is negative, the graph shifts DOWN by that many units.
  4. Every point (x, y) on f moves to (x, y + k).
Worked example
f(x) = x2 -> g(x) = x2 + 5
The +5 is added outside, after the function is evaluated: g(x) = f(x) + 5. Outside means same direction, so the parabola moves UP 5 units. Vertex: (0, 0) -> (0, 5).
Worked example
f(x) = |x| -> g(x) = |x| – 4
The -4 is outside: g(x) = f(x) – 4. Same direction, so the V shifts DOWN 4 units. Vertex: (0, 0) -> (0, -4). Check: g(0) = -4.
Watch out: Outside = same direction. g(x) = f(x) + 5 moves UP 5, never left or right — that is an inside change.

2Horizontal Shifts

Key idea: Changing x INSIDE the function, g(x) = f(x – h), moves the graph horizontally — but in the OPPOSITE direction of the sign: f(x – h) shifts right h, f(x + h) shifts left h.
  1. Find the change grouped with x, inside the parentheses.
  2. Rewrite it as (x – h) or (x + h).
  3. f(x – h) shifts RIGHT h units; f(x + h) shifts LEFT h units.
  4. Every point (x, y) on f moves to (x + h, y) for a right shift of h.
Worked example
f(x) = x2 -> g(x) = (x – 4)2
The -4 is inside, grouped with x. Inside means horizontal and OPPOSITE direction: minus means RIGHT. So the parabola shifts RIGHT 4 units. Check: g(0) = 16 and f(-4) = 16.
Worked example
f(x) = |x| -> g(x) = |x + 2|
The +2 is inside: |x + 2| = |x – (-2)|. Opposite direction: plus means LEFT. The V shifts LEFT 2 units. Check: g(0) = 2 and f(-2) = 2.
Watch out: The famous flip: (x – 4)^2 shifts RIGHT 4, even though the sign looks like minus. Inside = opposite direction, always.

3Reflecting Over the x-Axis

Key idea: A minus in FRONT of the function, g(x) = -f(x), reflects the graph over the x-axis: every point (x, y) becomes (x, -y).
  1. Spot the minus sign in front of the function (outside, not on x).
  2. Change the sign of every y-value: (x, y) -> (x, -y).
  3. Points on the x-axis (y = 0) do not move.
  4. The graph is now a mirror image across the x-axis.
Worked example
f(x) = x2 -> g(x) = -x2
The minus is in FRONT of the whole function: g(x) = -(x2). Every y-value changes sign: (1, 1) -> (1, -1). The parabola now opens downward. x-intercepts (like (0, 0)) stay put.
Worked example
f(x) = sqrt(x) -> g(x) = -sqrt(x)
g(x) = -f(x) flips the graph over the x-axis. The point (4, 2) on f becomes (4, -2) on g.
Watch out: -x^2 reflects the OUTPUT (flip over the x-axis). (-x)^2 is different — that minus is on x, so it reflects over the y-axis instead.

4Reflecting Over the y-Axis

Key idea: A minus attached to x INSIDE the function, g(x) = f(-x), reflects the graph over the y-axis: every point (x, y) becomes (-x, y).
  1. Spot the minus sign attached directly to x (inside the function).
  2. Change the sign of every x-value: (x, y) -> (-x, y).
  3. Points on the y-axis (x = 0) do not move.
  4. The graph is now a mirror image across the y-axis.
Worked example
f(x) = x3 -> g(x) = (-x)3
The minus is attached to x, inside: g(x) = f(-x). Every x-value changes sign: (2, 8) -> (-2, 8). The cubic flips left-to-right. The y-intercept stays put.
Worked example
f(x) = sqrt(x) -> g(x) = sqrt(-x)
g(x) = f(-x) reflects over the y-axis. The point (4, 2) on f becomes (-4, 2) on g, and the domain flips to x <= 0.
Watch out: -f(x) vs f(-x): a minus in FRONT flips over the x-axis; a minus on the x flips over the y-axis. Say which one out loud before you write.

5Vertical Stretches and Compressions

Key idea: Multiplying the whole function, g(x) = a * f(x), stretches or compresses vertically: |a| > 1 stretches by factor |a|, and 0 < |a| < 1 compresses by factor |a|. A negative a also reflects over the x-axis.
  1. Find the number a multiplying the whole function (outside).
  2. If |a| > 1: vertical stretch by a factor of |a| (y-values grow).
  3. If 0 < |a| < 1: vertical compression by a factor of |a| (y-values shrink).
  4. If a is negative, also reflect over the x-axis. Every point (x, y) -> (x, a*y).
Worked example
f(x) = |x| -> g(x) = 2|x|
The 2 multiplies the whole function (outside). |2| > 1, so it is a vertical stretch by a factor of 2: every y-value doubles. Point (1, 1) -> (1, 2). The V gets narrower.
Worked example
f(x) = x2 -> g(x) = 12x2
The 12 multiplies the whole function, and 0 < 1/2 < 1, so it is a vertical compression by a factor of 1/2: every y-value is halved. Point (2, 4) -> (2, 2). The parabola gets wider and flatter.
Watch out: Compression by 1/2 multiplies every y by 1/2 (graph gets flatter). Do not confuse it with a horizontal change — nothing happens to the x-values here.

6Horizontal Stretches and Compressions

Key idea: Multiplying x INSIDE, g(x) = f(b*x), stretches or compresses horizontally — with the inside/opposite twist: |b| > 1 compresses by factor 1/|b|, and 0 < |b| < 1 stretches by factor 1/|b|. A negative b also reflects over the y-axis.
  1. Find the number b multiplying x inside the function.
  2. If |b| > 1: horizontal compression by a factor of 1/|b| (x-values shrink).
  3. If 0 < |b| < 1: horizontal stretch by a factor of 1/|b| (x-values grow).
  4. If b is negative, also reflect over the y-axis. Every point (x, y) -> (x/b, y).
Worked example
f(x) = x2 -> g(x) = (2x)2
The 2 multiplies x inside. |2| > 1, so it is a horizontal compression by a factor of 1/2: every x-value is halved. Point (2, 4) -> (1, 4). Check: g(1) = (2)2 = 4.
Worked example
f(x) = |x| -> g(x) = |x/3|
x is multiplied by 1/3 inside, and 0 < 1/3 < 1, so it is a horizontal stretch by a factor of 3: every x-value triples. Point (3, 3) -> (9, 3). Check: g(9) = |9/3| = 3.
Watch out: Horizontal is the opposite twin: f(2x) squeezes the graph to HALF its width, not double. Inside changes always go against your first instinct.

7Order of Transformations

Key idea: When changes stack, list them in a fixed order: horizontal changes (shifts, stretches, reflections) first, then the outside multiplier, then the vertical shift. Apply them to points in that same order.
  1. Scan inside (with x): shifts, then stretches/compressions, then reflections.
  2. Scan outside: the multiplier (stretch + possible flip), then the added number (vertical shift).
  3. Apply inside changes to the x-coordinate and outside changes to the y-coordinate.
  4. Verify by plugging one point of f into g.
Worked example
f(x) = |x| -> g(x) = 2|x + 1| – 3
Handle one change at a time. Inside +1: shift LEFT 1. Outside 2: vertical stretch by 2. Outside -3: shift DOWN 3. Answer: shift left 1, vertical stretch by 2, shift down 3. Track a point: (-1, 1) -> x: -1 – 1 = -2; y: 2(1) – 3 = -1, so (-2, -1). Check: g(-2) = 2| -2 + 1| – 3 = -1.
Worked example
f(x) = x2 -> g(x) = -(x – 2)2 + 5
Inside -2: shift RIGHT 2. Front minus: reflect over the x-axis. Outside +5: shift UP 5. Answer: shift right 2, reflect over the x-axis, shift up 5. Check: g(2) = -(0)2 + 5 = 5.
Watch out: Order matters when a horizontal shift meets a horizontal stretch: 2(x – 1) is not the same graph as 2x – 1. Keep inside changes in the right sequence.

8Describing Transformations from Equations

Key idea: To describe any transformation, compare g to the parent: scan inside changes first (horizontal, opposite), then outside changes (vertical, same direction), and finish with a one-point check.
  1. Write down the parent function.
  2. List every inside change: (x – h) shifts, (bx) stretches, (-x) reflection.
  3. List every outside change: a*f(x) stretch/flip, +k vertical shift.
  4. State the full description and verify with one point.
Worked example
f(x) = sqrt(x) -> g(x) = -sqrt(x – 3)
Checklist: front minus -> reflect over the x-axis. Inside -3 -> shift RIGHT 3 (opposite direction). Nothing outside. Answer: reflect over the x-axis and shift right 3. Check: g(4) = -sqrt(1) = -1; f(1) = 1 flips to -1.
Worked example
f(x) = x3 -> g(x) = 13(-x + 2)3
Inside: -x + 2 = -(x – 2), so reflect over the y-axis and shift RIGHT 2. Outside 1/3: vertical compression by 1/3. Answer: reflect over the y-axis, shift right 2, vertical compression by 1/3.
Watch out: Do not mix inside and outside: in -sqrt(x) – 3, the front minus flips the graph but the -3 is OUTSIDE, so it shifts down 3 — not right 3.

9Writing Equations from Descriptions

Key idea: To write the equation, build it step by step: horizontal moves change x inside (opposite sign), reflections add a minus (front for x-axis, on x for y-axis), vertical moves add outside (same sign).
  1. Start with the parent function.
  2. Horizontal shifts: replace x with (x – h) for right h, (x + h) for left h.
  3. Reflections: put a minus in front for the x-axis, on the x for the y-axis.
  4. Stretches: multiply outside (a) or inside (b). Vertical shifts: add outside. Check with a point.
Worked example
Parent f(x) = x3; shift LEFT 1, UP 4
Left 1 is horizontal: change x inside with the opposite sign -> (x + 1)3. Up 4 is vertical: add outside with the same sign -> + 4. Answer: g(x) = (x + 1)3 + 4. Check: g(-1) = 4 = f(0) + 4.
Worked example
Parent f(x) = x2; shift RIGHT 2, DOWN 1, reflect over the x-axis
Right 2: inside, opposite sign -> (x – 2)2. Reflect over x-axis: front minus -> -(x – 2)2. Down 1: outside, same sign -> – 1. Answer: g(x) = -(x – 2)2 – 1. Check: g(2) = -1.
Watch out: Left 1 means (x + 1), not (x – 1). Say the opposite-sign rule out loud for every horizontal move — it is the #1 source of wrong equations.
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60-Second Challenge

How many function transformations problems can you solve in 60 seconds?