MAT 115 Worksheet: Graphing Polynomials without a Graphing Calculator
Things to refer to, to help you with graphing:
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Look at the leading term.
End behavior on far left and right will be the same direction as the graph of the leading term.
Symmetry:
If 𝑓(−𝑥) = 𝑓(𝑥), the graph will be symmetric to the y-axis.
If 𝑓(−𝑥) = −𝑓(𝑥), the graph will be symmetric to the origin.
Find the y-intercept, if there is one:
Let x = 0, and solve for y.
Find any x-intercepts:
Let y = 0. Factor the polynomial and solve for x.
Each x-intercept is also called a zero of the polynomial.
If the multiplicity of a zero is odd, the graph crosses the x-axis at that location.
If the multiplicity of a zero is even, the graph touches but does not cross the x-axis there.
For each of the following, determine the end behavior, any special symmetry, y-intercept, x-intercepts,
and behavior of the graph at each x-intercept. Then sketch a graph consistent with that information.
1. 𝑦 = 𝑥 4 − 25𝑥 2
2. 𝑦 = 𝑥 3 − 16𝑥 2
3. 𝑦 = 𝑥 3 − 16𝑥
4. 𝑦 = 𝑥 3 − 10𝑥 2 + 25𝑥
5. 𝑦 = 𝑥 4 − 13𝑥 2 + 36
6. 𝑦 = 𝑥 2 + 10𝑥 + 25
7. 𝑦 = 𝑥 2 + 8𝑥 + 12
8. 𝑦 = −𝑥 2 + 𝑥 + 6
9. 𝑦 = −𝑥 3 + 𝑥 2 + 6𝑥