The curve and its crossings
Read the curve of y = x² − 5x + 6. Where does it meet the x-axis?
First, move a different curve in the lab. Here k is how far the curve moves up. Count how many times it meets the x-axis.
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x from 0 to 5, y from 0 to 6.- A grid:
xfrom 0 to 5,yfrom 0 to 6. - Some points of
y = x² − 5x + 6. - The points lie on one smooth curve.
- The curve meets the axis at
x = 2. - It meets the axis again at
x = 3.
The rule
The curve meets the x-axis at x = 2 and x = 3. The x-axis is the horizontal line where y is 0.
The curve of a quadratic is a parabola. On the graph, the value is the height. The zeros are the x-values where the curve meets the x-axis. There the height is 0.
A parabola can meet the axis twice, once, or not at all. It cannot meet the axis three times. A quadratic has at most two zeros. So its curve meets the axis at most twice. A cubic is a polynomial with highest power x³. Its curve can meet the axis up to three times. The graph of ax + b, with a ≠ 0, meets the axis once.
- One grid for three quadratics.
y = x² − 1crosses twice: at −1 and 1.y = x²touches once, at 0.y = x² + 1never crosses.
In the lab, k = 3, k = 4 and k = 5 draw these three curves. At k = 4 the curve touches the axis once.
Sometimes a curve touches the axis. It meets the axis once and turns back. A touch is one zero. When you add or multiply the zeros, count that zero twice. For example, x² is (x − 0)(x − 0), so the zero 0 appears twice.
The curve y = x² − 5 has no whole-number zero. It still has two zeros: √5 and −√5. The curve crosses the axis between whole numbers.