Geometrical meaning of the zeroes
The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis.
Reading zeroes from a graph
Draw the graph of y = p(x). At every point where it meets the x-axis, y = 0. So the x-coordinate of that point is a zero of p(x). Count the meeting points to get the number of zeroes. If the graph only touches the axis at a point, count that point once. The graph of a quadratic is a U-shaped curve called a parabola.
Three cases for a quadratic y = ax² + bx + c
| What the parabola does | Number of zeroes | Example |
|---|---|---|
| Cuts the x-axis at two points | 2 | y = x² − 4 (zeroes −2 and 2) |
| Touches the x-axis at one point | 1 (two equal zeroes) | y = x² − 6x + 9 (zero 3) |
| Does not meet the x-axis | 0 | y = x² + 1 |
How to draw the graph of y = x² − 2x − 3
- 1Make a table of values. For x = −2, −1, 0, 1, 2, 3, 4 you get y = 5, 0, −3, −4, −3, 0, 5.
- 2Draw the x-axis and y-axis on graph paper. Choose a scale such as 1 cm = 1 unit and write it in a corner. Label O, x and y.
- 3Plot the points (−2, 5), (−1, 0), (0, −3), (1, −4), (2, −3), (3, 0) and (4, 5).
- 4Join them with a smooth curve, not with straight lines. It is a parabola opening upward, because a = 1 > 0.
- 5★Mark and label the points where the curve cuts the x-axis: (−1, 0) and (3, 0). Write 'zeroes: −1 and 3'.
★ key step
Example 1: read the zeroes from a description
The graph of y = p(x) cuts the x-axis at (−4, 0), (1, 0) and (5, 0), and nowhere else. How many zeroes does p(x) have? What are they?
- Each point where the graph meets the x-axis gives one zero.
- There are 3 such points.
- The zeroes are their x-coordinates: −4, 1 and 5.
p(x) has 3 zeroes: −4, 1 and 5.
Example 2: check the graph by factorising
Check that the zeroes of x² − 2x − 3 read from the graph above are correct.
- Split the middle term: two numbers with product −3 and sum −2 are −3 and 1.
- x² − 2x − 3 = x² − 3x + x − 3
- = x(x − 3) + 1(x − 3) = (x − 3)(x + 1)
- x − 3 = 0 gives x = 3; x + 1 = 0 gives x = −1.
The zeroes are 3 and −1, the same as on the graph.
The path of water from a fountain, or of a ball hit for a six, is shaped like a parabola. The points where the path meets the ground are like the zeroes of a quadratic polynomial.
1 marks: “The graph of y = p(x) touches the x-axis at one point and does not cross it. How many zeroes does p(x) have?” One zero (two equal zeroes), because the graph meets the x-axis at only one point.
1 marks: “The graph of y = ax² + bx + c opens downward. What is the sign of a?” a < 0 (a is negative).
2 marks: “The graph of y = p(x) is a parabola that cuts the x-axis at (−3, 0) and (2, 0). Find p(x) if its x² coefficient is 1.” The zeroes are −3 and 2, so p(x) = (x + 3)(x − 2) = x² + x − 6.
Zeroes live on the x-axis: wherever the graph touches or crosses it, read off x.