Why This Matters

The SAT uses the same transformation language across almost every major function family.

Quadratics use it.

Absolute value functions use it.

Exponentials use it.

Logs use it.

Radicals use it.

Cubics use it.

Instead of memorizing a separate rule for every graph, learn one pattern:

a · f(x - h) + k

such that f is the specific parent function you are referencing.

This is the SAT's universal graph transformation language.

The Universal Form

Most transformed functions can be written as:

y = a · f(x - h) + k

Think of this as:

Parent Function + Instructions

What Does h Mean?

Horizontal Shift

The graph moves:

• Right h units if x - h
• Left h units if x + h

The Weird Rule

Students expect:

x + 3

to move right.

It does not.

It moves left.

Always remember:

Opposite Direction Inside

Quick Checkpoint

Which graph is shifted 5 units right?

A) y = (x + 5)²

B) y = (x - 5)²

C) y = x² + 5

D) y = x² - 5

Answer

B

Inside = opposite.

What Does k Mean?

Vertical Shift

The graph moves:

• Up k units if +k
• Down k units if -k

This one behaves normally.

Quick Checkpoint

Which graph is shifted 8 units down?

A) y = (x - 2)² - 8

B) y = (x + 2)² + 8

C) y = (x - 8)²

D) y = x² + 8

Answer

A

Outside = same direction.

The Most Important SAT Idea

The point (h,k) is usually an important point on the graph.

The exact meaning depends on the function family.

The SAT often asks about this point directly.

Quadratics

Parent:

Transformed:

y = a(x - h)² + k

What Does (h,k) Represent?

The vertex.

The highest or lowest point.

SAT Clues

• maximum
• minimum
• vertex
• turning point

Immediately look for:

(h,k)

Example

y = (x - 4)² + 7

Vertex:

(4, 7)

Checkpoint

What is the vertex?

y = -2(x + 3)² - 5

Answer

(-3, -5)

Absolute Value

Parent:

y = |x|

Transformed:

y = a|x - h| + k

What Does (h,k) Represent?

The vertex.

The point of the V.

Example

y = |x - 6| + 2

Vertex:

(6, 2)

SAT Clue

The graph has a sharp corner.

Look for:

(h,k)

Radical Functions

Parent:

y = √x

Transformed:

y = a√(x - h) + k

What Does (h,k) Represent?

The starting point.

The graph begins here.

Example

y = √(x - 9) + 4

Starting point:

(9, 4)

SAT Clue

Questions about:

• domain
• where the graph begins
• smallest x-value

Exponential Functions

Parent:

y = $2^x$

Transformed:

y = a · $2^{(x-h)}$+ k

What Does k Represent?

The horizontal asymptote.

Not the point.

The asymptote.

Example

y = $2^{(x-3) }$+ 5

Horizontal asymptote:

y = 5

SAT Clues

• growth
• decay
• asymptote
• long-term behavior

Checkpoint

What is the horizontal asymptote?

y = 4( $3^x$) - 7

Answer

y = -7

Logarithmic Functions

Parent:

y = log(x)

Transformed:

y = log(x - h) + k

What Does h Represent?

The vertical asymptote.

Example

y = log(x - 4) + 2

Vertical asymptote:

x = 4

SAT Clues

• domain
• asymptote
• logarithmic graph

Checkpoint

What is the vertical asymptote?

y = log(x + 6)

Answer

x = -6

Cubic Functions

Parent:

y = x³

Transformed:

y = a(x - h)³ + k

What Does (h,k) Represent?

The inflection point.

The middle of the S-curve.

Example

y = (x - 2)³ + 5

Inflection point:

(2, 5)

What Does “a” Mean?

The SAT also tests the coefficient a.

If a Is Positive

The graph keeps its normal orientation.

If a Is Negative

The graph reflects.

Examples:

Quadratic:

• Positive → opens up
• Negative → opens down

Absolute Value:

• Positive → V
• Negative → upside-down V

Exponential:

• Positive → growth shape
• Negative → reflected growth shape

If |a| > 1

Vertical stretch.

Graph becomes steeper.

If 0 < |a| < 1

Vertical compression.

Graph becomes wider or flatter.

SAT Recognition Drill

Identify the important feature being displayed.

1

y = (x - 8)² + 3

What point is being shown?

2

y = |x + 2| - 6

What point is being shown?

3

y = √(x - 7) + 1

Where does the graph begin?

4

y = 5 · $2^x$ - 4

What is the horizontal asymptote?

5

y = log(x - 9)

What is the vertical asymptote?

6

y = -(x - 3)² + 8

Does the parabola open upward or downward?

Answers

1

Vertex = (8, 3)

2

Vertex = (-2, -6)

3

Starting point = (7, 1)

4

Horizontal asymptote:

y = -4

5

Vertical asymptote:

x = 9

6

Downward

The negative a reflects the parabola.

SAT Takeaway

When you see:

a· f(x - h) + k

do not immediately start graphing.

Ask:

1. What function family is this?
2. What does (h,k) represent for that family?
3. What does a do?

Most SAT graph questions can be answered from those three questions alone.