Reading and Interpreting Functions
- qian58
- May 10
- 2 min read
Updated: May 17
Function problems on the Regents are mostly about reading carefully and applying slope-intercept form correctly.
The Regents tests functions a few consistent ways: identifying whether a relation is a function, interpreting the parts of a linear equation in context, and reading transformations of parabolas. Once you know what to look for, these are reliable points.
Click here for tips on reading and interpreting functions
The essentials
A relation is not a function if any x-value appears more than once with a different y-value.
In y = mx + b: m is the rate of change per unit; b is the starting value when x = 0.
Transformations of y = x²:
(x + 3)² shifts the graph left 3 (sign is opposite the direction
(x − 3)² shifts it right 3
x² − 2 shifts it down 2
⚠Left/right shifts feel backwards. (x + 3)² moves the vertex left to x = −3. Quick check: set the inside equal to zero — x + 3 = 0 gives x = −3. That's where the new vertex sits.
Worked Example
Problem: Marcus is saving money. He starts with $40 and saves $15 every week. The total amount he has saved after x weeks is modeled by y = 40 + 15x.
(a) How much money did Marcus start with? (b) How much does he save each week? (c) How much will he have after 6 weeks?
Try it yourself! Below is a real Regents problem from January 2024
Problem:The Speedy Jet Ski Rental Company models its total cost as y = 30 + 40x, where x is hours rented and y is total cost in dollars.
(a) What is the insurance fee? (b) What is the hourly rental rate? (c) What is the total cost for a 3-hour rental?
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STEP-BY-STEP SOLUTION
Match to slope-intercept form y = mx + b. Rewriting: y = 40x + 30, so m = 40 and b = 30.
(a) Insurance fee = b = $30 — the flat fee before any hours.
(b) Hourly rental rate = m = $40 per hour.
(c) Substitute x = 3:
y = 30 + 40 * 3 = 30 + 120 = $150
Insurance fee: $30. Hourly rate: $40/hr. 3-hour rental: $150.
✗ COMMON TRAP
Swapping slope and intercept — calling $40 the fee and $30 the rate. The constant is always the flat fee; the coefficient of x is always the per-unit rate.
✓ QUICK CHECK
What does the company charge for 0 hours? That's the flat fee — and it's the constant, since 40 × 0 = 0.
For more practice with systems of functions, check out these resources:
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