1 markMCQSection A

The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is : (a) 0 (b) 1 (c) 2 (d) 4

Q: 4. The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is : (a) 0 (b) 1 (c) 2 (d) 4

(c) 2 — the graph cuts the x-axis at exactly two points, so $f(x)$ has 2 zeroes.

Polynomials

Number of zeroes from graph

Options

(A)0

(B)1

(D)4

Official Answer

zeroes of a polynomialgraph cuts x-axisnumber of intersection pointsgeometrical meaning of zeroy = f(x)two zeroesx-intercepts

Marking Scheme

11 mark: correct option (c) 2, based on the graph cutting the x-axis at two points.
2Acceptable if a differently-printed figure clearly shows a different number of x-axis intersections; the marking follows the actual crossings shown.

Hint

Count the number of points where the curve meets (cuts or touches) the x-axis — that is the number of zeroes.

Quick Oral Answer

The number of zeroes is the number of points where the graph meets the x-axis; here it crosses at two points, so $f(x)$ has 2 zeroes.

Analysis & Explanation

This MCQ tests the geometrical meaning of zeroes of a polynomial from its graph.

Concept

The number of zeroes of $y = f(x)$ equals the number of points where the graph cuts or touches the x-axis.

Key points

The given curve meets the x-axis at 2 distinct points, so $f(x)$ has 2 zeroes.
0 zeroes would mean the graph never touches the axis; 1 would mean a single touch/cut; 4 would need four crossings.

Common mistakes

Counting the number of "humps" in the curve instead of actual x-axis intersections.
Note: the exact count depends on the printed figure — always count the actual crossings shown.

Common Mistakes

1Counting where the curve meets the y-axis instead of the x-axis — zeroes correspond only to x-axis intersections.
2Counting the number of turning points (peaks/valleys) rather than x-axis crossings.
3Assuming the number of zeroes equals the degree without actually looking at how many times the graph meets the x-axis.

Interesting Facts

A polynomial of degree n can have AT MOST n real zeroes, so it can cut the x-axis at most n times — but it may have fewer if some zeroes are complex or repeated.

When a graph just TOUCHES the x-axis and turns back (does not cross), that point is a repeated (double) zero — it counts once as a location but represents two equal roots.

The idea of reading zeroes off a graph connects algebra to coordinate geometry, a link formalised by René Descartes in the 17th century through the Cartesian plane.

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Frequently Asked Questions

How do you find the number of zeroes of a polynomial from its graph?

Count the number of distinct points where the graph of $y = f(x)$ cuts or touches the x-axis. Each such point is a zero, so the number of x-axis intersections equals the number of real zeroes.

Does a graph touching the x-axis count as a zero?

Yes. A point where the curve touches the x-axis and turns back is a repeated (double) zero. It is one location on the graph but corresponds to two equal roots of the polynomial.

Why is the answer here 2 and not 4?

The shown curve meets the x-axis at only two points. For the answer to be 4, the graph would have to cross the x-axis at four separate points. Since only two crossings are visible, $f(x)$ has 2 zeroes.