Identify and interpret intercepts (especially the y-intercept). | Linear functions | SAT Math

Intercepts tell you where the line crosses the axes. The y-intercept is the starting value.

Core Idea

The y-intercept is the output value when the input is zero — it's the starting value. The x-intercept is the input value when the output is zero.

Understanding

Intercepts answer two natural questions: Where does the function start? When does the output hit zero?

The y-intercept is the value of $𝑦$ when $𝑥 = 0$ . In $𝑦 = 𝑚 𝑥 + 𝑏$ , it's $𝑏$ . In a word problem, it usually represents an initial amount, a fixed fee, or a starting condition — something that exists before the input variable kicks in.

The x-intercept is where $𝑦 = 0$ . Set the equation equal to zero and solve for $𝑥$ . In context, this often means "when does the quantity run out?" or "when does the balance reach zero?"

On the SAT, y-intercept interpretation questions are far more common than x-intercept ones. The typical trap: confusing the intercept with the slope, or picking an answer that describes the slope's meaning instead.

Step by Step

For the y-intercept: set $𝑥 = 0$ and solve for $𝑦$ , or read $𝑏$ directly from $𝑦 = 𝑚 𝑥 + 𝑏$ .
For the x-intercept: set $𝑦 = 0$ and solve for $𝑥$ .
Interpret in context: the y-intercept is the starting value (before any change occurs); the x-intercept is when the output reaches zero.

Misconceptions

Thinking the y-intercept is a rate — it's a fixed starting value, not a per-unit change.
On a graph, reading the x-intercept as the y-intercept (mixing up which axis the line crosses).
Forgetting that the y-intercept can be negative — a debt of $200 at $𝑡 = 0$ means $𝑏 = - 200$ .

Question

Worked Example

A store offers a gift card with an initial balance of $50. Each purchase uses $8 from the card. The remaining balance $𝐵$ , in dollars, after $𝑝$ purchases is $𝐵 = 50 - 8 𝑝$ . What does the 50 represent in this equation?

Select an answer to see the explanation