Interpreting the Y-Intercept in Context
The y-intercept is the value when the input is zero — the starting amount, before anything changes.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
In $C = 15t + 40$, setting $t = 0$ gives $C = 40$: the cost before any time passes. That constant term is the initial value, not a rate.
How to solve one, step by step
Example: in $C = 15t + 40$, what does $40$ represent?
- Set $t = 0$: $C = 40$.
- So $40$ is the cost at the very start — a fixed, one-time amount.
The mistakes that cost points
- Calling the intercept a rate. The intercept is a starting value; the slope is the rate.
- Reading the x-intercept instead. The y-intercept is where the input is zero, not where the output is zero.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
The total cost in dollars ($y$) of a taxi ride covering $x$ miles is given by $y = 4x + 8$. Select the correct interpretation of the equation.
- AThe taxi charges a total of $\$12$ per mile.
- BThe taxi ride costs a total of $\$12$.
- CThe taxi charges $\$4$ per mile and there is an initial flat fee of $\$8$.
- DThe taxi charges $\$8$ per mile and there is an initial flat fee of $\$4$.
Show solution
Answer: C, The taxi charges $\$4$ per mile and there is an initial flat fee of $\$8$.. In the equation $y = mx + b$, the slope $m$ is the rate of change (cost per mile) and the y-intercept $b$ is the initial value (the cost when $x = 0$). Here, $m = 4$ and $b = 8$, so the taxi charges $\$4$ per mile with an initial flat fee of $\$8$.
Medium
The number of pages $P$ remaining in a book after $h$ hours of reading is modeled by $P = -40h + 320$. Which of the following statements is true?
- AThe reader reads $320$ pages per hour and starts with $40$ pages remaining.
- BThe reader reads at $320$ pages per hour throughout.
- CThe reader started with $320$ pages and reads at a rate of $40$ pages per hour.
- DThe reader finishes the book after $320$ hours of reading.
Show solution
Answer: C, The reader started with $320$ pages and reads at a rate of $40$ pages per hour.. In $P = -40h + 320$, the slope is $-40$ pages per hour (the reading rate) and the $P$-intercept is $320$. When $h = 0$, $P = 320$, so the book has $320$ pages at the start. The rate of reading is the slope, $40$ pages per hour. The intercept is the initial page count, not the rate.
Hard
A small business tracks its monthly profit. In January, the business had an existing debt of $\$500$. Each month it earns a net profit of $\$200$. The monthly profit-loss balance $B$, in dollars, after $m$ months is modeled by a linear equation. If the balance after $3$ months is $\$100$, which of the following equations correctly models $B$ in terms of $m$, and what does the $y$-intercept represent?
- A$B = 200m - 500$; the $y$-intercept represents the number of months until the business breaks even.
- B$B = -500m + 200$; the $y$-intercept $200$ represents the initial profit before debts are applied.
- C$B = 200m - 500$; the $y$-intercept $-500$ represents the initial debt of $\$500$ before any months of profit.
- D$B = 200m - 500$; the $y$-intercept $-500$ represents the monthly loss rate of $\$500$.
Show solution
Answer: C, $B = 200m - 500$; the $y$-intercept $-500$ represents the initial debt of $\$500$ before any months of profit.. The business starts at $-\$500$ (debt) and gains $\$200$ per month, so $B = 200m - 500$. Check: at $m = 3$, $B = 200(3) - 500 = 600 - 500 = 100$. ✓ The $y$-intercept is $-500$, meaning when $m = 0$ (before any months), the balance is $-\$500$. This is the initial condition, not a rate.
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