Transformations of Functions — SAT Math Guide

Q: What You Need to Know

• **Horizontal shifts**: f(x + c) moves left c units, f(x - c) moves right c units • **Vertical shifts**: f(x) + d moves up d units, f(x) - d moves down d units • **Reflections**: -f(x) flips over x-axis, f(-x) flips over y-axis • **Vertical stretch/compression**: af(x) stretches by factor |a| if

Transformations of functions SAT questions test your ability to understand how graphs shift, flip, and stretch when you modify a function's equation. These problems involve taking a parent function like f(x) and applying changes to create g(x) = af(bx + c) + d. The Advanced Math domain includes 4-6 questions on function transformations, making this a crucial topic for boosting your Digital SAT score. You'll find these concepts more intuitive once you see the patterns.

What You Need to Know

Horizontal shifts: f(x + c) moves left c units, f(x - c) moves right c units

Vertical shifts: f(x) + d moves up d units, f(x) - d moves down d units

Reflections: -f(x) flips over x-axis, f(-x) flips over y-axis

Vertical stretch/compression: af(x) stretches by factor |a| if |a| > 1, compresses if 0 < |a| < 1

Horizontal stretch/compression: f(bx) compresses by factor 1/|b| if |b| > 1, stretches if 0 < |b| < 1

📐 KEY FORMULA: g(x) = af(b(x - h)) + k where (h,k) is the new center point

💡 PRO TIP: Remember "opposite day" for horizontal shifts — f(x + 3) actually moves LEFT 3 units!

How to Solve Transformations of Functions SAT Questions

Example Question 1 — Medium Difficulty

The graph of y = f(x) passes through the point (2, 5). If g(x) = f(x - 3) + 4, which point must be on the graph of y = g(x)?

A) (-1, 9)

B) (5, 9)

C) (5, 1)

D) (-1, 1)

Solution:

Step 1: Identify the transformation — f(x - 3) shifts right 3 units, +4 shifts up 4 units

Step 2: Apply horizontal shift to x-coordinate: 2 + 3 = 5

Step 3: Apply vertical shift to y-coordinate: 5 + 4 = 9

✅Answer: B — The point (2, 5) becomes (5, 9) after shifting right 3 and up 4.

Example Question 2 — Hard Difficulty

The function h(x) = -2f(3x + 6) - 1 is a transformation of f(x). Compared to the graph of f(x), the graph of h(x) is:

A) Reflected over the x-axis, compressed horizontally by factor 1/3, shifted left 2 units, and shifted down 1 unit

B) Reflected over the x-axis, stretched vertically by factor 2, compressed horizontally by factor 1/3, shifted left 2 units, and shifted down 1 unit

C) Reflected over the y-axis, compressed horizontally by factor 1/3, shifted right 2 units, and shifted up 1 unit

D) Stretched vertically by factor 2, compressed horizontally by factor 3, shifted left 6 units, and shifted down 1 unit

Solution:

Step 1: Rewrite in standard form: h(x) = -2f(3(x + 2)) - 1

Step 2: Identify each transformation: -2 means reflect over x-axis and stretch vertically by 2

Step 3: 3(x + 2) means compress horizontally by 1/3 and shift left 2 units

Step 4: -1 means shift down 1 unit

✅Answer: B — All transformations: reflection over x-axis, vertical stretch by 2, horizontal compression by 1/3, left 2, down 1.

Common SAT Math Mistakes to Avoid

❌Mistake: Confusing horizontal shift direction (thinking f(x + 3) moves right)

✅Fix: Remember f(x - h) moves right h units, f(x + h) moves left h units

❌Mistake: Mixing up vertical stretch factor with horizontal compression factor

✅Fix: The number multiplying f(x) affects vertical changes, the number multiplying x affects horizontal changes

❌Mistake: Forgetting that negative signs create reflections

✅Fix: -f(x) flips over x-axis, f(-x) flips over y-axis — know both types

❌Mistake: Not factoring out coefficients before identifying transformations

✅Fix: Always rewrite f(bx + c) as f(b(x + c/b)) to see the true horizontal shift

Practice Question — Try It Yourself

The function g(x) = 3f(2x) - 5 is a transformation of f(x). If the point (4, 7) lies on the graph of y = f(x), which point lies on the graph of y = g(x)?

A) (2, 16)

B) (8, 16)

C) (2, 2)

D) (8, 2)

Show Answer

Answer: A — The point (4, 7) becomes (2, 16). Horizontal compression by 1/2 changes x from 4 to 2, vertical stretch by 3 and shift down 5 changes y from 7 to 3(7) - 5 = 16.

Key Takeaways for the SAT

Master the "opposite day" rule for horizontal shifts to avoid careless errors on SAT math transformations of functions

Always factor out coefficients first: f(2x + 6) = f(2(x + 3)) shows the shift clearly

Negative signs mean reflections — don't miss this on the Digital SAT

Vertical transformations affect y-values, horizontal transformations affect x-values

Practice identifying multiple transformations in one function for College Board's harder questions