The key idea
In y=a(x−h)2+k, the vertex is (h,k) and the extreme value is k. The vertex sits halfway between any two inputs with equal outputs, including the two zeros, so h=2r+s for y=a(x−r)(x−s).
The trap
Reading the vertex of a(x+3)2 as x=3; the sign flips.
Can Desmos do it?
Desmos marks the vertex of a graph you can type in. When a coefficient is unknown, use symmetry to find h first; then one more fact pins down the rest.
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(x)=x2+14x+c, where c is a constant, has a minimum value of −7. What is the value of c?
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Answer: 42The minimum occurs at x=−214=−7, where f=49−98+c=c−49. Setting c−49=−7 gives c=42.
Level 2 · Test level
The function f is defined by f(x)=a(x+3)(x−7), where a is a constant. The maximum value of f is 25. What is the value of f(7)?
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Answer: 0The vertex is halfway between the zeros −3 and 7, at x=2. So f(2)=a(5)(−5)=−25a=25, which gives a=−1. Then f(7)=−1(10)(0)=0.
Level 3 · Beyond the test
The height, in feet, of a ball t seconds after it is thrown is h(t)=−16t2+bt+7, where b is a constant. The ball is at the same height at t=1 and at t=5. What is the maximum height, in feet, that the ball reaches?
Show the answer
Answer: 151Equal heights put the vertex halfway between the two times, at t=3. The vertex of −16t2+bt+7 is at t=32b, so b=96. The maximum is h(3)=−16(3)2+96(3)+7=151.
Practice all 36 questions
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