Absolute Value Functions — SAT Math Guide

Q: What You Need to Know

• Absolute value of x equals the distance from x to zero on the number line • The basic function f(x) = |x| creates a V-shaped graph with vertex at origin (0,0) • Inside the absolute value bars, negative outputs become positive • Transformations follow the pattern f(x) = a|x - h| + k where (h,k) is

Absolute value functions SAT questions test your understanding of how these V-shaped graphs behave and transform. These functions always output non-negative values, creating distinctive graphs that can shift, stretch, or flip depending on their equation form. You'll encounter 2-3 absolute value function problems in the SAT math section's Advanced Math domain. Master these concepts and you'll confidently tackle any absolute value challenge the Digital SAT throws your way.

What You Need to Know

Absolute value of x equals the distance from x to zero on the number line

The basic function f(x) = |x| creates a V-shaped graph with vertex at origin (0,0)

Inside the absolute value bars, negative outputs become positive

Transformations follow the pattern f(x) = a|x - h| + k where (h,k) is the vertex

When a is negative, the graph flips upside down (opens downward)

When |a| > 1, the graph stretches vertically; when 0 < |a| < 1, it compresses

The domain is all real numbers; the range depends on transformations

📐 KEY FORMULA: f(x) = a|x - h| + k (vertex form)

💡 PRO TIP: The vertex of f(x) = a|x - h| + k is always at point (h, k), not (-h, k)!

How to Solve Absolute Value Functions on the SAT

Example Question 1 — Medium Difficulty

The function f(x) = 2|x - 3| + 1 has its vertex at which point?

A) (-3, 1)

B) (3, -1)

C) (3, 1)

D) (-3, -1)

Solution:

Step 1: Identify the vertex form f(x) = a|x - h| + k

Step 2: Compare with f(x) = 2|x - 3| + 1 to find h = 3 and k = 1

Step 3: The vertex is at (h, k) = (3, 1)

✅Answer: C — The vertex form places the vertex at (h, k), which is (3, 1).

Example Question 2 — Hard Difficulty

If g(x) = -½|x + 4| - 2, which statement about the graph of g(x) is true?

A) The graph opens upward with vertex at (-4, -2)

B) The graph opens downward with vertex at (4, -2)

C) The graph opens downward with vertex at (-4, -2)

D) The graph opens upward with vertex at (-4, 2)

Solution:

Step 1: Rewrite as g(x) = -½|x - (-4)| + (-2) to identify a = -½, h = -4, k = -2

Step 2: Since a = -½ is negative, the graph opens downward

Step 3: The vertex is at (h, k) = (-4, -2)

Step 4: Confirm the coefficient ½ compresses the graph vertically

✅Answer: C — Negative coefficient means downward opening, and vertex is at (-4, -2).

Common SAT Math Mistakes to Avoid

❌Mistake: Confusing the vertex location in f(x) = a|x - h| + k as (-h, k)

✅Fix: Remember the vertex is always at (h, k), so if you see |x - 3|, the vertex x-coordinate is +3

❌Mistake: Thinking absolute value functions can have negative outputs

✅Fix: The expression inside the bars can be negative, but |anything| ≥ 0 always

❌Mistake: Forgetting that negative coefficients flip the graph upside down

✅Fix: When a < 0 in f(x) = a|x - h| + k, the V-shape opens downward like an upside-down V

❌Mistake: Mixing up domain and range for transformed absolute value functions

✅Fix: Domain is always all real numbers; range depends on the vertex and direction of opening

Practice Question — Try It Yourself

The function h(x) = 3|x - 1| - 4 is graphed on the coordinate plane. What is the minimum value of h(x)?

A) -4

B) -1

C) 1

D) 3

Show Answer

Answer: A — The vertex is at (1, -4), and since the coefficient 3 is positive, the graph opens upward, making -4 the minimum value.

Key Takeaways for the SAT

Master the vertex form f(x) = a|x - h| + k where vertex is at (h, k)

Negative coefficients flip the graph to open downward

SAT math absolute value functions often test transformations and vertex identification

Always check if the graph opens up or down to determine minimum or maximum values

Use the vertex to quickly identify key features without extensive graphing