What is the Y-intercept of a Function?

S3-SA5-0027

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Y-intercept of a function is the point where its graph crosses the Y-axis. At this point, the value of x is always zero, and the Y-intercept tells us the starting value or initial condition of the function.

Simple Example

Quick Example

Imagine you start saving money for a new cricket bat. You already have 50 rupees in your piggy bank. This 50 rupees is your starting amount, even before you add any new money. In a graph showing your savings over time, 50 rupees would be your Y-intercept, representing your savings when time (x) is zero.

Worked Example

Step-by-Step

Let's find the Y-intercept for the function y = 3x + 7.

Step 1: Remember that the Y-intercept occurs when x = 0.

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Step 2: Substitute x = 0 into the function's equation.

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Step 3: y = 3(0) + 7

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Step 4: y = 0 + 7

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Step 5: y = 7

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Step 6: The Y-intercept is the point (0, 7).

Answer: The Y-intercept is (0, 7).

Why It Matters

Understanding the Y-intercept helps us know the starting point of any process, like the initial battery charge of your phone or the base price of a taxi ride. Engineers use it to design systems, and data scientists use it to understand initial trends in data. It's crucial for careers in AI/ML, data science, and even economics.

Common Mistakes

MISTAKE: Confusing the Y-intercept with the X-intercept. | CORRECTION: The Y-intercept is where the graph crosses the Y-axis (x=0), while the X-intercept is where it crosses the X-axis (y=0).

MISTAKE: Forgetting to substitute x=0. | CORRECTION: Always set x=0 in the function's equation to find the corresponding y-value, which is the Y-intercept.

MISTAKE: Writing only the y-value as the intercept. | CORRECTION: The Y-intercept is a point, so it should be written as an ordered pair (0, y-value), not just 'y-value'.

Practice Questions

Try It Yourself

QUESTION: What is the Y-intercept of the function y = 2x - 5? | ANSWER: (0, -5)

QUESTION: A function is given by f(x) = 4 - x. Find its Y-intercept. | ANSWER: (0, 4)

QUESTION: The cost of an auto-rickshaw ride is given by C = 15 + 8d, where C is the total cost and d is the distance in kilometers. What does the Y-intercept represent in this scenario? | ANSWER: The Y-intercept is (0, 15). It represents the fixed base fare of 15 rupees, even if you travel 0 km.

MCQ

Quick Quiz

For the function y = -x + 10, what is its Y-intercept?

(10, 0)

(0, 10)

(0, -10)

(-10, 0)

The Correct Answer Is:

The Y-intercept occurs when x=0. Substituting x=0 into y = -x + 10 gives y = -(0) + 10, so y = 10. Thus, the Y-intercept is (0, 10).

Real World Connection

In the Real World

When you buy mobile data, some plans have a fixed monthly charge even before you use any data. This fixed charge is like the Y-intercept in a cost function. Similarly, in a taxi app, the base fare that you pay even for a short distance is the Y-intercept, representing the cost when the distance traveled (x) is zero.

Key Vocabulary

Key Terms

Y-axis: The vertical line in a coordinate plane | X-axis: The horizontal line in a coordinate plane | Ordered Pair: A set of two numbers (x, y) that show the position of a point on a graph | Function: A rule that assigns exactly one output for each input | Equation: A mathematical statement that two expressions are equal

What's Next

What to Learn Next

Great job understanding the Y-intercept! Next, you should explore the X-intercept. Knowing both X and Y intercepts will give you a complete picture of where a function's graph crosses the axes, which is super useful for plotting and analyzing graphs!