The graph of a quadratic polynomial y = p(x) cuts the x-axis at (−2, 0) and (3, 0), and cuts the y-axis at (0, −6). Find the zeroes of p(x) and find p(x).
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. The graph of y = p(x) meets the x-axis at (−2, 0) and (3, 0), and the y-axis at (0, −6). To find: the zeroes of p(x), and p(x).
2 · Write the rule. The zeroes of p(x) are the x-coordinates of the points where the graph meets the x-axis. If the zeroes are α and β, then p(x) = k(x − α)(x − β).
3 · Read the zeroes. The graph meets the x-axis where x = −2 and x = 3. So the zeroes are −2 and 3.
4 · Write p(x) with k. p(x) = k(x + 2)(x − 3) = k(x² − x − 6)
5 · Use the point on the y-axis. At (0, −6): p(0) = k(0 − 0 − 6) = −6k. So −6k = −6 and k = 1.
6 · Check and box the answer. p(x) = x² − x − 6 = (x − 3)(x + 2), which is 0 at x = 3 and x = −2, and p(0) = −6. Therefore the zeroes are −2 and 3, and p(x) = x² − x − 6.
Scoring tip: Say in words that the zeroes are the x-coordinates of the points on the x-axis. Then use the third point to find k.
