Intercepts and Their Meaning in Context

3.4: Intercepts and Their Meaning in Context. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 4 of 63 parts Practice Flashcards

1/3

Starting value

Objective: Interpret a y-intercept in context.

Prerequisite skills: linear equations and interpreting coordinate pairs. Setting x = 0 gives the model output before any input change. The intercept may represent an initial amount, a fixed fee, or a baseline, depending on how variables are defined. Common error: using the slope as the starting amount. Restrictions and interpretation: The x-intercept is (a,0), while the y-intercept is (0,b); consider feasible input values.

Formulas

$y-intercept = f(0)$

Worked examples

Try itA delivery model is C = 25d + 33, where C is dollars and d is distance in miles. What is the fixed charge in dollars?Show answer

At d = 0, the distance charge vanishes and C = 33. Thus the fixed fee is $33. Therefore the requested answer is 33.

Swipe sideways to move between sections

Starting value

Objective: Interpret a y-intercept in context.

Formulas

$y-intercept = f(0)$

Worked examples

Try itA delivery model is C = 25d + 33, where C is dollars and d is distance in miles. What is the fixed charge in dollars?Show answer

At d = 0, the distance charge vanishes and C = 33. Thus the fixed fee is $33. Therefore the requested answer is 33.

Zero output

Objective: Find an x-intercept and attach its meaning.

Set the output equal to zero to locate an x-intercept. In a depletion model, this can be the time when a quantity is exhausted. A graph beyond that point might be algebraically defined but physically inappropriate. Common error: reporting the initial quantity rather than the depletion time. Restrictions and interpretation: The x-intercept is (a,0), while the y-intercept is (0,b); consider feasible input values.

Formulas

$mx + b = 0 ⇒ x = −b/m, m ≠ 0$

Worked examples

Try itA tank follows V = 648 − 24t, where V is liters and t is minutes. At what time is the tank empty?Show answer

Set V = 0: 0 = 648 − 24t. Add 24t and divide by 24: t = 648/24 = 27 minutes. Therefore the requested answer is 27.

Both intercepts in a model

Objective: Relate a total constraint to its axis intercepts.

In a resource equation ax + by = C, the intercept C/a gives the all-x allocation and C/b gives the all-y allocation. Both variables must obey the contextual restrictions, often nonnegative values. Common error: mixing the coefficient of one variable with the other intercept. Restrictions and interpretation: The x-intercept is (a,0), while the y-intercept is (0,b); consider feasible input values. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$ax + by = C ⇒ x-intercept C/a and y-intercept C/b$

Worked examples

Try itA budget equation is 3x + 5y = 375. If x and y are nonnegative quantities, what is the greatest possible y?Show answer

To maximize y under the nonnegative constraint, set x = 0. Then 5y = 375, giving y = 75. Any positive x uses budget and lowers y. Therefore the requested answer is 75.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonFinding Equations from Points, Tables, and Graphs Next lessonParallel and Perpendicular Lines