Concept 2.2
Zeroes: Where the Graph Meets the x-axis
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
A real number k is a zero of a polynomial p(x) if p(k) = 0. Geometrically, the zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis.
The intuition
A zero is where the output is zero, and on a graph the output is the height. So zeroes sit exactly where the curve touches or crosses the x-axis.
y = x² − 2x − 3 crosses at x = −1 and x = 3, so those are its zeroes. Its meeting point with the y-axis, (0, −3), is just its value at x = 0: a different question.
The attention
Look at where the graph meets the x-axis.
y = x² − 2x − 3 meets the x-axis at (−1, 0) and (3, 0), the y-axis at (0, −3)
Look hereNot here
- Look here
- Where the graph meets the x-axis. On the x-axis, y = 0. So the x-coordinates there are exactly the values that make the polynomial zero.
- Not here
- Where it meets the y-axis. The y-intercept is the value of the polynomial at x = 0. It is easy to spot, but it answers a different question.
- What it teaches
- A graph answers many questions. Ask which line on the graph your question lives on.