Question: Find the $ y $-intercept of the line that passes through the points $ (1, 3) $ and $ (4, 9) $. - Coaching Toolbox

Understanding the Context

Why the $ y $-Intercept Matters in Everyday Life

Right now, more people than ever are relying on clear data insights in daily decision-making—whether tracking monthly savings, analyzing employment growth, or understanding performance curves in education or income. The $ y $-intercept reveals the baseline value: what’s expected when input is zero. For example, a startup might estimate user count at launch (y-intercept) before growth trends take hold (slope). Similarly, a student analyzing salary progression might use this intercept as a starting benchmark. This concept is not abstract—it shapes forecasting, planning, and informed choices.

Understanding how to calculate this intercept demystifies linear patterns embedded in real-world data, empowering users to engage confidently with graphs, tables, and digital tools used widely in US workplaces, schools, and public forums.

Key Insights

How to Find the $ y $-Intercept: A Clear, Step-by-Step Approach

The $ y $-intercept is found using two key points and a basic formula from linear algebra. For the line through $ (1, 3) $ and $ (4, 9) $, start by calculating the slope: rise over run.

Slope $ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2 $.
Then use point-slope form $ y - y_1 = m(x - x_1) $:

$ y - 3 = 2(x - 1) $
Simplify: $ y = 2x - 2 + 3 $
So $ y = 2x + 1 $

From this equation, the $ y $-intercept is the constant term: $ (0, 1) $. Alternatively, plug either point into $ y = mx + b $ to solve for $ b $. Using $ (1, 3) $:
$ 3 = 2(1) + b \Rightarrow b = 1 $

Final Thoughts

Thus, the $ y $-intercept is $ (0, 1)