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What is the X-intercept of a Function?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The X-intercept of a function is the point where the graph of the function crosses or touches the X-axis. At this point, the value of the Y-coordinate is always zero. It tells us the input value (x) for which the output value (y) is zero.
Simple Example
Quick Example
Imagine a graph showing how many runs a cricket team needs to score (Y-axis) versus the overs remaining (X-axis). The X-intercept would be the point where the runs needed become zero, meaning the team has won! It shows the number of overs remaining when the target is achieved.
Worked Example
Step-by-Step
Let's find the X-intercept of the function: y = 2x - 6.
1. To find the X-intercept, we know that the Y-coordinate is always 0. So, we set y = 0.
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2. Substitute y = 0 into the equation: 0 = 2x - 6.
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3. Now, we need to solve this equation for x. Add 6 to both sides: 0 + 6 = 2x - 6 + 6.
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4. This simplifies to: 6 = 2x.
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5. Divide both sides by 2 to isolate x: 6 / 2 = 2x / 2.
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6. So, x = 3.
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7. The X-intercept is the point where x = 3 and y = 0.
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Answer: The X-intercept is (3, 0).
Why It Matters
Understanding X-intercepts helps scientists and engineers find 'break-even points' or 'equilibrium points' in various systems. For example, in economics, it helps find when profit is zero, or in physics, when an object returns to its starting position. Data scientists use it to understand when a certain condition is met.
Common Mistakes
MISTAKE: Setting x = 0 to find the X-intercept. | CORRECTION: To find the X-intercept, you must set the Y-coordinate (y) to 0, not x.
MISTAKE: Writing the X-intercept as just a number (e.g., '3'). | CORRECTION: The X-intercept is a point on the graph, so it should always be written as an ordered pair (x, 0), for example, (3, 0).
MISTAKE: Confusing X-intercept with Y-intercept. | CORRECTION: X-intercept is where the graph crosses the X-axis (y=0). Y-intercept is where it crosses the Y-axis (x=0). They are different points.
Practice Questions
Try It Yourself
QUESTION: Find the X-intercept of the function y = 3x - 9. | ANSWER: (3, 0)
QUESTION: A function is given by f(x) = 5x + 15. What is its X-intercept? | ANSWER: (-3, 0)
QUESTION: The cost of producing 'x' samosas is given by C(x) = 10x - 50. If the revenue is also 10x, find the X-intercept of the profit function P(x) = Revenue - Cost. | ANSWER: (5, 0)
MCQ
Quick Quiz
For a function, what is true at the X-intercept?
The x-coordinate is zero.
The y-coordinate is zero.
Both x and y coordinates are zero.
The slope of the line is zero.
The Correct Answer Is:
B
At the X-intercept, the graph crosses the X-axis. Any point on the X-axis has a y-coordinate of zero. The x-coordinate can be any real number.
Real World Connection
In the Real World
Imagine a startup company in Bengaluru tracking its monthly profit. If they plot a graph of 'profit vs. number of products sold', the X-intercept would show the 'break-even point' – the number of products they need to sell to make zero profit (meaning no loss, no gain). This helps them plan their sales targets.
Key Vocabulary
Key Terms
X-AXIS: The horizontal number line in a coordinate plane | Y-AXIS: The vertical number line in a coordinate plane | COORDINATE PLANE: A two-dimensional surface formed by two intersecting number lines | ORDERED PAIR: A pair of numbers (x, y) that represent a point on a coordinate plane | FUNCTION: A rule that assigns exactly one output to each input
What's Next
What to Learn Next
Great job understanding X-intercepts! Next, you should learn about the Y-intercept of a function. It's the 'sister concept' to the X-intercept and together, they help you sketch graphs of linear functions quickly and easily.
