Math · College algebra · Worked example
Find the maximum of a quadratic function
A ball is thrown upward from 1.5 m above the ground, and its height after t seconds is h(t) = −4.9t² + 19.6t + 1.5 meters. When does it reach its greatest height, and how high is that?
Recognize the shape
The coefficient of t² is −4.9 < 0, so the graph of h opens downward and its vertex is the highest point.
Find the time of the vertex
Use t = −b/(2a) with a = −4.9 and b = 19.6.
Find the greatest height
Substitute t = 2.
Check with symmetry
The parabola is symmetric about t = 2, so the ball is back at its starting height, 1.5 m, at t = 4.
State the model
The model ignores air resistance, which is reasonable for a short throw of a small, heavy ball.
Result
The ball is highest at t = 2 s, 21.1 m above the ground.
Your turn
At a price of p dollars, a shop sells 400 − 20p shirts a week. Which price gives the greatest weekly revenue, and how much is it?
Show the answer and explanation
$10, for $2,000 a week.
Revenue is R(p) = p(400 − 20p) = −20p² + 400p. Its vertex is at p = −400/(2(−20)) = 10, and R(10) = 10 × 200 = 2,000.
Keep exploring
Graph plots the height with the peak and the two moments at 1.5 m marked.
Return to the concept →Sources and scope
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