Why This Matters
Many SAT Math questions are not really testing algebra.
They are testing whether you can recognize what type of function you are looking at.
When you identify the function family correctly, you immediately gain information about:
- shape
- intercepts
- asymptotes
- maximums and minimums
- end behavior
- what Desmos should look like
Think of parent functions like recognizing animal species.
You don't need to know everything about the animal.
You just need to recognize it quickly.
The SAT Recognition Game
When you see a graph, equation, table, or word problem, ask:
What family does this belong to?
The SAT repeatedly tests the same handful of families.
Parent Function #1: Linear
Parent Equation
y = x
What It Looks Like
- Straight line
- Constant slope
- Constant rate of change
SAT Clues
You may see:
- "increases by 5 each year"
- "constant rate"
- "slope"
- "unit rate"
Instant Recognition
If the output changes by the same amount every time:
x | y |
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
The change is always +3.
This is linear.
Checkpoint 1
A company's profits increase by $4,000 every year.
What type of function is most likely being modeled?
A) Linear
B) Exponential
C) Quadratic
D) Radical
Answer
A) Linear
The increase is constant.
Parent Function #2: Quadratic
Parent Equation
y = x²
What It Looks Like
- U-shape
- One highest or lowest point
- Symmetry
SAT Clues
You may see:
- maximum value
- minimum value
- projectile motion
- area optimization
- vertex
Instant Recognition
Outputs do not increase by a constant amount.
Instead the increases keep changing.
x | y |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
Checkpoint 2
A ball is launched upward and eventually falls back to the ground.
Which function family most commonly models its height?
A) Quadratic
B) Linear
C) Exponential
D) Logarithmic
Answer
A) Quadratic
Projectile motion is usually quadratic.
Parent Function #3: Absolute Value
Parent Equation
y = |x|
What It Looks Like
- V-shape
- Sharp corner
- Symmetry
SAT Clues
You may see:
- distance from a point
- absolute difference
- V-shaped graph
Instant Recognition
If the graph has a sharp point instead of a smooth curve:
Think:
Absolute Value
Checkpoint 3
A graph decreases linearly until x = 3 and then immediately begins increasing linearly.
What family is most likely represented?
A) Quadratic
B) Absolute Value
C) Cubic
D) Exponential
Answer
B) Absolute Value
The sharp turning point is the giveaway.
Parent Function #4: Exponential
Parent Equation
y =
What It Looks Like
- Slow growth
- Rapid growth later
- Horizontal asymptote
SAT Clues
You may see:
- percent growth
- percent decay
- doubles every year
- halves every year
- compound interest
Instant Recognition
The change is multiplicative.
Example:
x | y |
1 | 2 |
2 | 4 |
3 | 8 |
4 | 16 |
The outputs are multiplied, not added.
Checkpoint 4
A bacteria population doubles every 12 hours.
Which family best models this situation?
A) Linear
B) Quadratic
C) Exponential
D) Radical
Answer
C) Exponential
Doubling is exponential growth.
Parent Function #5: Logarithmic
Parent Equation
y = log(x)
What It Looks Like
- Opposite of exponential growth
- Vertical asymptote
- Rapid growth early
- Slow growth later
SAT Clues
You may see:
- pH scale
- Richter scale
- decibels
- logarithms
Instant Recognition
If you see:
log(x)
You are looking at a logarithmic function.
Checkpoint 5
Which function family contains a vertical asymptote and grows more slowly as x increases?
A) Linear
B) Logarithmic
C) Quadratic
D) Absolute Value
Answer
B) Logarithmic
Parent Function #6: Radical
Parent Equation
y = √x
What It Looks Like
- Starts somewhere
- Curves right
- Cannot continue indefinitely left
SAT Clues
You may see:
- square roots
- domains
- starting points
Instant Recognition
The graph usually begins at a specific point and extends in one direction.
Checkpoint 6
Which parent function usually has a clearly visible starting point?
A) Linear
B) Radical
C) Quadratic
D) Exponential
Answer
B) Radical
Parent Function #7: Rational
Parent Equation
y = 1/x
What It Looks Like
- Two disconnected branches
- Vertical asymptote
- Horizontal asymptote
SAT Clues
You may see:
- denominator contains a variable
- asymptotes
- excluded values
Instant Recognition
Variable in the denominator?
Immediately suspect a rational function.
Checkpoint 7
Which family is most likely being represented?
y = (x + 2)/(x - 5)
A) Quadratic
B) Rational
C) Exponential
D) Radical
Answer
B) Rational
SAT Recognition Drill
Identify the function family without solving.
1
A graph has one maximum value and is symmetric.
2
A quantity increases by 7 every month.
3
A graph has a vertical asymptote and a variable in the denominator.
4
A quantity triples every year.
5
A graph forms a V shape.
6
A graph begins at a point and extends only to the right.
7
A graph has a horizontal asymptote and doubles repeatedly.
8
A graph has two disconnected branches approaching asymptotes.
Answers
1
Quadratic
2
Linear
3
Rational
4
Exponential
5
Absolute Value
6
Radical
7
Exponential
8
Rational
SAT Takeaway
Before solving any function question, ask:
What family am I looking at?
Recognizing the parent function often tells you more than the equation itself.
The SAT rewards recognition far more often than it rewards memorization.