What are the Roots of a Function from its Graph?

S3-SA5-0304

Grade Level:

Class 9

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

Definition

What is it?

The roots of a function, when seen on its graph, are the special points where the graph crosses or touches the X-axis. At these points, the value of the function (which is represented by the Y-coordinate) is exactly zero.

Simple Example

Quick Example

Imagine a graph showing how many runs a cricket team scored in each over. If the graph dips down and touches the X-axis at 'Over 10', it means the team scored 0 runs in that specific over. This 'Over 10' point on the X-axis would be a 'root' of that run-scoring function.

Worked Example

Step-by-Step

Let's find the roots of the function shown by a graph that looks like a 'U' shape (a parabola).

1. Look at the graph carefully.
2. Find the horizontal line, which is the X-axis.
3. Identify all the points where the 'U' shaped graph intersects (crosses) the X-axis.
4. Suppose the graph crosses the X-axis at X = -2.
5. And it also crosses the X-axis at X = 3.
6. These X-values, -2 and 3, are the roots of the function.

Answer: The roots of the function from the graph are X = -2 and X = 3.

Why It Matters

Understanding roots helps scientists and engineers predict outcomes, like when a rocket launched by ISRO will hit the ground, or when a company's profit will be zero. Data scientists use roots to find critical points in their data, which helps them make better decisions for apps like Zomato or Swiggy.

Common Mistakes

MISTAKE: Confusing roots with Y-intercepts. Students sometimes look at where the graph crosses the Y-axis. | CORRECTION: Roots are ONLY the X-values where the graph crosses or touches the X-axis (where Y=0). The Y-intercept is where it crosses the Y-axis (where X=0).

MISTAKE: Giving the full coordinate (x, y) as the root. For example, saying '(-2, 0)' is a root. | CORRECTION: The root is only the X-value. So, if the graph crosses the X-axis at (-2, 0), the root is X = -2, not the entire coordinate.

MISTAKE: Missing roots if the graph only touches the X-axis instead of crossing it. | CORRECTION: Even if the graph just 'bounces off' or touches the X-axis at a single point without crossing, that point is still a root. The Y-value is still zero there.

Practice Questions

Try It Yourself

QUESTION: A straight line graph passes through the points (0, 4) and (2, 0). What is the root of this function? | ANSWER: X = 2

QUESTION: A graph of a function touches the X-axis at X = -5 and crosses the X-axis at X = 10. List all the roots of this function. | ANSWER: X = -5 and X = 10

QUESTION: Imagine a graph of a ball thrown upwards. It starts at (0, 0), goes up, and then comes down, hitting the ground (X-axis) at X = 8. What are the roots of the function representing the ball's path? | ANSWER: X = 0 and X = 8

MCQ

Quick Quiz

Which of the following describes the roots of a function from its graph?

The points where the graph crosses the Y-axis.

The highest or lowest points of the graph.

The points where the graph crosses or touches the X-axis.

The starting point of the graph.

The Correct Answer Is:

Option C is correct because roots are defined as the X-values where the function's output (Y-value) is zero, which means the graph intersects the X-axis. Options A, B, and D describe other features of a graph, not its roots.

Real World Connection

In the Real World

In economics, companies often graph their profit over time. If the profit graph dips to touch the X-axis (where profit is zero), those points are roots. These roots tell the company when they 'break even' – neither making a profit nor a loss. This is crucial for planning new business ventures or pricing products like a new mobile phone model.

Key Vocabulary

Key Terms

ROOTS: The X-values where a function's graph crosses or touches the X-axis (Y=0). | X-AXIS: The horizontal number line on a graph. | Y-AXIS: The vertical number line on a graph. | GRAPH: A visual representation of a function. | FUNCTION: A rule that assigns exactly one output for each input.

What's Next

What to Learn Next

Now that you understand roots from a graph, you can learn about finding roots of functions using algebraic methods, like factorization or the quadratic formula. This will help you find roots even when you don't have a graph!