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Master Function Transformations (The Graph Shortcut)

Master Function Transformations (The Graph Shortcut)

This is one of the biggest SAT speed boosts: instead of re-graphing from scratch,
you start with a parent function and apply quick moves (shift, stretch, reflect).

Big idea:
Most SAT graph questions are just:
start with parent
apply transformations
read key points

If you can do transformations cleanly, you can match graphs to equations in seconds.

Quick Jump (click to jump)
Learn the moves + avoid the classic traps
1) Parent Functions (Your Starting Shapes)

The 5 shapes SAT uses most
 quadraticabsradical
2) The 6 Transformations (Move List)

Shift, reflect, stretch, compress
 shiftreflectstretch
3) Inside vs Outside (The #1 SAT Trap)

Why x-h shifts right and +k shifts up
 insideoutsidetrap
4) Transform Key Points (Fast Graphing Method)

Use anchor points instead of drawing perfectly
 pointsanchors
5) Common SAT Traps

Horizontal moves, negative signs, scaling
 signsorderscales
6) Practice

Build speed with transformation drills
 drillsspeed
7) SAT-Style Multiple Choice

Match equation ↔ graph changes
 MCQmatch
8) Comparison Table (Cheat Sheet)

Move → what it does → example
 fast lookuptest-day

Start: Parent Functions
Jump to MCQ
Jump to Cheat Sheet
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Parent Functions (Your Starting Shapes)

SAT transformations are easiest when you memorize a few “parent graphs.”
You don’t need perfect drawings — just recognize the shape.

Quadratic

parabola

f(x)=x^2

U-shape with vertex at (0,0).

Absolute Value

V-shape

f(x)=|x|

Corner at (0,0), symmetric left/right.

Square Root

endpoint

f(x)=\u221Ax

Starts at (0,0), only defined for x ≥ 0.

Reciprocal

hyperbola

f(x)=1/x

Two branches, asymptotes at x=0 and y=0.

Exponential

growth/decay

f(x)=b^x

Passes through (0,1), approaches y=0 for decay bases.

The 6 Transformations (Move List)

These show up everywhere. Learn them once and reuse forever.

Shifts (Translations)

Up / Down

g(x)=f(x)+k

Up if k>0, down if k<0.

Right / Left

g(x)=f(x-h)

Right if h>0, left if h<0.

Reflections

Over x-axis

g(x)=-f(x)

Flips the graph vertically.

Over y-axis

g(x)=f(-x)

Flips the graph horizontally.

Stretches / Compressions

Vertical stretch/compress

g(x)=a\cdot f(x)

|a|>1 stretches, 0<|a|<1 compresses, a<0 also reflects over x-axis.

Horizontal stretch/compress

g(x)=f(bx)

|b|>1 compresses, 0<|b|<1 stretches, b<0 also reflects over y-axis.

Quick Example (All-in-One)

Start: f(x)=x^2

g(x)= -2(x-3)^2 + 5
  • Right 3
  • Up 5
  • Reflect over x-axis
  • Vertical stretch by 2

Inside vs Outside (The #1 SAT Trap)

If it’s inside the function (with x), it affects horizontal movement/scale.
If it’s outside, it affects vertical movement/scale.

Outside Changes y

f(x)+k → up/down

a·f(x) → vertical stretch/compress

-f(x) → reflect over x-axis

Think “outside messes with outputs.”

Inside Changes x (and feels “backwards”)

f(x-h) → right by h

f(x+h) → left by h

f(bx) → horizontal compress if |b|>1

Inside changes act on inputs, so they “flip” your instinct.

One-liner to memorize:
Inside changes go the opposite direction for shifts, and opposite for stretches too.

Transform Key Points (Fast Graphing Method)

You don’t need to draw the whole graph. Pick a few anchor points on the parent,
transform the points, and you basically have the graph.

Example with a Quadratic

Parent: f(x)=x^2 key points:

(0,0), (1,1), (-1,1), (2,4), (-2,4)

Transform: g(x)=(x-3)^2+2

  • Right 3: add 3 to x-coordinates
  • Up 2: add 2 to y-coordinates
(3,2), (4,3), (2,3), (5,6), (1,6)

Example with Absolute Value

Parent: f(x)=|x| key points:

(0,0), (1,1), (-1,1), (2,2), (-2,2)

Transform: g(x)=-|x+2|+1

  • Left 2: x becomes -2 at the corner
  • Reflect over x-axis (negative sign)
  • Up 1
Corner: (-2,1) and opens downward

Student move:
If the SAT shows you a graph, find the vertex/corner/endpoint first.
Then identify shifts and reflections from there.

Common SAT Traps (and How to Avoid Them)

  • Shift direction inside parentheses: f(x-3) is right 3 (not left).
  • Horizontal scale confusion: f(2x) compresses (not stretches).
  • Negative inside vs outside: f(-x) reflects over y-axis; -f(x) reflects over x-axis.
  • Order panic: you can apply transformations step-by-step. Don’t try to do it all at once.

Trap Example

Compare g(x)=|x-4| and h(x)=|x+4|.
The corners are at x=4 and x=-4 — inside signs are opposite of what your brain wants to do.

Practice: Transformation Drills

Don’t solve. Just describe the moves and key points.

Set A: Describe the Transformations

  1. g(x)=(x+2)^2-7 from f(x)=x^2
  2. h(x)=-|x-3|+4 from f(x)=|x|
  3. p(x)=\u221A(x-5)+1 from f(x)=\u221Ax
  4. q(x)=2^x-6 from f(x)=2^x
  5. r(x)=1/(x+1) from f(x)=1/x

Set B: Find the Key Feature Fast

  1. Vertex of -(x-1)^2+3
  2. Corner of |x+5|-2
  3. Endpoint of \u221A(x-4)
  4. Horizontal asymptote of 3(0.8)^x+7
  5. Vertical asymptote of 1/(x-9)

Training goal:
Start every transformation question by finding the “anchor”:
vertex (quadratic), corner (abs), endpoint (radical), asymptote (rational/exponential).

SAT-Style Multiple Choice (Transformations)

These are classic “match the graph” transformation questions.

Question 1

The graph of y=x^2 is transformed to the graph of
y=(x-3)^2+5. Which describes the transformation?

  1. Left 3 and up 5
  2. Right 3 and up 5
  3. Right 5 and up 3
  4. Left 5 and up 3

Question 2

Which function is a reflection of f(x)=|x| over the x-axis?

  1. |x|+2
  2. -|x|
  3. |x-2|
  4. | -x |

Question 3

The graph of y=\u221Ax is transformed to
y=\u221A(x+4). What happened?

  1. Shift right 4
  2. Shift left 4
  3. Shift up 4
  4. Shift down 4

Question 4

Compared to f(x)=x^2, what does g(x)=f(2x) do?

  1. Horizontal stretch by factor 2
  2. Horizontal compression by factor 2
  3. Vertical stretch by factor 2
  4. Vertical compression by factor 2

Question 5

The function h(x)=-(x+1)^2 is created from f(x)=x^2.
Which two transformations occurred?

  1. Right 1, reflect over y-axis
  2. Left 1, reflect over x-axis
  3. Up 1, reflect over x-axis
  4. Down 1, reflect over y-axis

Speed move:
On MCQ graph questions, find the vertex/corner/endpoint first.
The location of that anchor usually eliminates 2–3 answer choices immediately.

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Cheat Sheet: Transformation → Effect → Example

Use this as your quick reference while practicing.

Transformation What It Does Example
Up/Down f(x)+k Shift up if k>0, down if k<0 x^2+3 is up 3
Right/Left f(x-h) Shift right by h (x-4)^2 is right 4
Reflect over x-axis -f(x) Flip vertically -|x|
Reflect over y-axis f(-x) Flip horizontally (-x)^2=x^2 (quadratic unchanged)
Vertical scale a f(x) Stretch if |a|>1, compress if 0<|a|<1 2x^2 is steeper
Horizontal scale f(bx) Compress if |b|>1, stretch if 0<|b|<1 (2x)^2=4x^2 compresses horizontally
Final habit:
Identify the parent, locate the anchor (vertex/corner/endpoint), then apply shifts and reflections.
Don’t overdraw — transform a few points and you’re done.

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