ACT Preparation / Unit 11 ACT 1022 worked example, 60 to 90 seconds
Reading Zeros From a Graph
The idea
To write a polynomial from its graph, work in four steps. First, read each x-intercept. Second, decide each one's multiplicity: crossing suggests odd, often 1; touching and turning suggests even, often 2. Third, write the factors, x minus each zero, to its multiplicity. Fourth, find the leading coefficient a from one more point, often the y-intercept, and check the end behavior.
In practice
Worked example: a graph crosses the x-axis at −2, touches it at 1, and passes through (0, 4). Factors: (x + 2)(x − 1)². At x = 0: a(2)(1) = 4, so a = 2. The polynomial is f(x) = 2(x + 2)(x − 1)², of degree 3 with a positive leading coefficient, so it falls on the left and rises on the right.
Where people slip
Check the degree against the ends: an odd degree must have opposite ends. If the graph's ends disagree, a multiplicity is wrong.
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