medium difficulty5 min

Sign pattern and graph interpretation

The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.

In one line: Above is positive, below is negative, on is zero.

Practice This Concept

Understand it

Graphs and signs work together in polynomial study.
If the graph is above the x-axis, the polynomial values are positive there.
If it is below the x-axis, the values are negative.
Where the graph meets the x-axis, the value is zero.
This helps students answer interpretation questions quickly without full algebraic solving.

Example

If a graph crosses the x-axis twice, the polynomial has two real zeros.

Formula and rule

Above x-axis implies p(x)>0, on x-axis implies p(x)=0, and below x-axis implies p(x)<0.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1If a graph crosses the x-axis at x=1 and x=3, then the polynomial is zero at those points.
2Between and outside those points, the sign may change depending on the graph's direction.

Method you can reuse

1Find the x-intercepts of the graph.
2Mark regions above and below the x-axis.
3Read positive and negative intervals.
4State the zeros at the intercepts.

Visual tip

A rough sketch should show x-intercepts, regions above and below the axis, and the corresponding sign of the polynomial.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does it mean when a polynomial graph is above the x-axis?

Think first, then show answer

It means the polynomial values are positive there. Above the x-axis indicates positive values, on the x-axis indicates zero, and below the x-axis indicates negative values.

Common wrong answer

A common wrong answer is to treat every x-intercept as a turning point. A graph may cross the axis or just touch it, so the movement matters.

Don't mix this up

Students sometimes think every polynomial must have the same sign on both sides of the x-axis. The sign can change when the graph crosses the axis.

Exam tip

Read the graph from left to right and note where it is above, on, or below the x-axis.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Interpret a graph, identify zeros, state intervals of positivity or negativity, or connect graph shape with algebraic values.

1
Enough for 1 mark
The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.
2
Add for 2 marks
Example: If a graph crosses the x-axis twice, the polynomial has two real zeros.
3
Add for 3 marks
Graphs and signs work together in polynomial study.

MCQ Quiz

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