The key idea
f(x−h) moves the graph h units right, f(x)+k moves it up k, and f(cx) squeezes it horizontally by a factor of c, so the input that made f zero must now be c times smaller. Track a key point, like the vertex or a zero, through each change.
The trap
Moving left for f(x−h), or multiplying zeros by c for f(cx) instead of dividing.
Can Desmos do it?
If f is given as a formula, graph g and look. If f is described only by its zeros or vertex, reason about the shifts.
College Board skill: Nonlinear functions · Advanced Math
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
The function f is defined by f(x)=x2−2x+11. The function g is defined by g(x)=f(x−k)+m, where k and m are constants. The vertex of the graph of y=g(x) is (8,8). What is the value of k+m?
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Answer: 5Complete the square: f(x)=(x−1)2+10, so the vertex of f is (1,10). The graph of g is shifted k right and m up, so its vertex is (1+k,10+m)=(8,8). Then k=7, m=−2, and k+m=5.
Level 2 · Test level
The zeros of the function f are −2 and 3, and f has no other zeros. The function g is defined by g(x)=f(x−3). What is the sum of the zeros of g?
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Answer: 7g(x)=0 when x−3=−2 or x−3=3, so the zeros of g are 1 and 6, with sum 7. The graph moved 3 units right.
Level 3 · Beyond the test
The zeros of the function f are −8 and 10, and f has no other zeros. The function g is defined by g(x)=f(21x)+1. What is the sum of all values of x for which g(x)=1?
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Answer: 4g(x)=1 exactly when f(21x)=0, that is, when 21x=−8 or 21x=10. So x=−16 or x=20, with sum 4. Replacing x with 21x doubles each zero of f.
Practice all 36 questions
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