Where algebra meets a graph
The graph of y=p(x) turns an algebraic zero into a visible point: y is zero exactly on the x-axis.
Learning goals
- Explain why a zero is the x-coordinate of an x-axis intersection.
- Count and identify zeroes from supplied graphs.
- Infer a simplest polynomial from its graph zeroes and degree.
Concept lesson
Axes and points
Every point on the x-axis has y-coordinate 0. A point (k,0) on y=p(x) therefore says p(k)=0.
Zero means x-axis meeting
On y=p(x), a zero is an x-coordinate where the curve cuts or touches the x-axis. Do not read the y-axis intercept and do not count bends.
Degree sets a maximum
A degree n polynomial can have at most n real zeroes. A linear polynomial has one; a quadratic may have 0, 1 or 2; a cubic may have 0, 1, 2 or 3. 'At most' does not mean 'exactly'.
Read zeroes from a supplied graph
A curve meets the x-axis at (-2,0), (1,0) and (4,0). Write its zeroes.
Quadratic touching once
A parabola touches the x-axis at x=3 and turns back. It has one distinct real zero, 3, even though its degree is 2.
Infer a simplest polynomial
A quadratic graph crosses at x=-1 and x=3. Find a simplest polynomial.
Intersections, not bends
On the interactive graph, mark every point where y=0, then list only its x-coordinate. The checker separately records any y-intercept or turning-point selection.
Possible zero counts
For degrees 1, 2 and 3, select every possible number of distinct real zeroes. Explain why degree gives an upper bound.
Diagnose a graph-reading error
A student reports two zeroes because the curve has two turning points, although it crosses the x-axis three times. Correct the method and the answer.
One- and three-mark graph patterns
For one mark, list x-coordinates of x-axis meetings. For a linked three-mark item, state degree, list zeroes, write factors and expand the simplest polynomial.
Exit ticket
From a supplied quadratic graph crossing at -3 and 2: list zeroes, explain why the y-intercept is irrelevant, state the maximum possible zero count, and form a simplest polynomial.