What is the Graph of y = mx + c?

S3-SA5-0100

Grade Level:

Class 9

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

Definition

What is it?

The graph of y = mx + c is a straight line. It shows how two quantities, 'x' and 'y', are related in a simple, constant way. This equation is called the 'slope-intercept form' of a linear equation.

Simple Example

Quick Example

Imagine you pay 10 rupees for an auto-rickshaw ride (fixed charge) and then 5 rupees for every kilometer you travel. Here, 'c' is 10 (fixed charge), 'm' is 5 (cost per km). If 'x' is kilometers and 'y' is total cost, the equation is y = 5x + 10. The graph would be a straight line showing total cost increasing steadily with distance.

Worked Example

Step-by-Step

Let's plot the graph of y = 2x + 1.

Step 1: Choose some simple values for 'x'. Let's pick x = -2, 0, 1, 2.
---
Step 2: Calculate the corresponding 'y' values using the equation.
If x = -2, y = 2(-2) + 1 = -4 + 1 = -3. So, point is (-2, -3).
---
Step 3: If x = 0, y = 2(0) + 1 = 0 + 1 = 1. So, point is (0, 1).
---
Step 4: If x = 1, y = 2(1) + 1 = 2 + 1 = 3. So, point is (1, 3).
---
Step 5: If x = 2, y = 2(2) + 1 = 4 + 1 = 5. So, point is (2, 5).
---
Step 6: Plot these points (-2, -3), (0, 1), (1, 3), (2, 5) on a graph paper.
---
Step 7: Draw a straight line passing through all these points. This is the graph of y = 2x + 1.

Answer: The graph is a straight line passing through the calculated points.

Why It Matters

Understanding y = mx + c is crucial because it helps predict trends and relationships in data. Data scientists use it to build models, physicists use it to describe motion, and economists use it to understand market behavior. It's foundational for fields like AI/ML and engineering.

Common Mistakes

MISTAKE: Confusing 'm' and 'c' values, especially when the equation isn't written as y = mx + c. | CORRECTION: Always rearrange the equation to the y = mx + c form first to clearly identify 'm' (slope) and 'c' (y-intercept).

MISTAKE: Plotting points incorrectly by mixing up x and y coordinates (e.g., plotting (y, x) instead of (x, y)). | CORRECTION: Remember that the first number in a coordinate pair is always the x-value (horizontal position) and the second is the y-value (vertical position).

MISTAKE: Not extending the line through the plotted points, making it look like a line segment. | CORRECTION: A linear graph represents ALL possible solutions to the equation, so always draw arrows at both ends of the line to show it continues infinitely.

Practice Questions

Try It Yourself

QUESTION: What is the y-intercept of the line y = 3x - 5? | ANSWER: -5

QUESTION: Plot any three points for the equation y = -x + 2. | ANSWER: Possible points: (0, 2), (1, 1), (2, 0). (Any three correct points are acceptable)

QUESTION: A mobile data plan costs Rs. 100 as a base charge, plus Rs. 50 for every GB used. Write the equation in the form y = mx + c and find the total cost if 3 GB of data is used. | ANSWER: Equation: y = 50x + 100. For 3 GB, y = 50(3) + 100 = 150 + 100 = Rs. 250.

MCQ

Quick Quiz

Which of these statements is true about the graph of y = mx + c?

It is always a curve.

It always passes through the origin (0,0).

It is always a straight line.

The value of 'm' is always positive.

The Correct Answer Is:

The equation y = mx + c is the standard form for a linear equation, which always produces a straight line graph. It doesn't always pass through the origin unless c=0, and 'm' can be positive or negative.

Real World Connection

In the Real World

Many real-world situations can be modeled by y = mx + c. For example, a delivery app like Swiggy or Zomato might have a base delivery charge (c) plus a charge per kilometer (m) for longer distances. The total delivery cost (y) for a certain distance (x) would follow this linear relationship.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation whose graph is a straight line. | SLOPE: The 'm' in y = mx + c, representing the steepness or rate of change of the line. | Y-INTERCEPT: The 'c' in y = mx + c, representing the point where the line crosses the y-axis (when x=0). | COORDINATE PLANE: A 2D surface defined by x and y axes where points are plotted.

What's Next

What to Learn Next

Great job understanding linear graphs! Next, you should explore how to find the slope and y-intercept directly from a given graph. This will help you write the equation y = mx + c just by looking at a line!