Maximum and Minimum in Context

In this lesson you will: Build a quadratic model from a description and find its maximum or minimum; Say which quantity each coordinate of the vertex measures; Restrict the domain to the inputs the situation allows.

The vertex answers two different questions

A ball is thrown upward from a platform 80 feet high at 64 feet per second. Its height after t seconds is h(t)=-16t^2+64t+80.

The axis is t=-(64)/(2(-16))=2, and h(2)=-64+128+80=144. Those two numbers answer different questions. The 2 says when the ball is highest, in seconds. The 144 says how high it gets, in feet. Confusing them is the single most expensive error in this lesson.

Products that pull in two directions

A club sells shirts at $10 each and sells 60 of them. A survey says every $1 added to the price costs them 5 sales. Revenue is price times quantity, and both of those change together.

Let x be the number of dollar increases. Revenue is R(x)=(10+x)(60-5x)=-5x^2+10x+600. One factor rises while the other falls, and their product is a quadratic that peaks in the middle. The axis is x=-(10)/(2(-5))=1, so the best price is $11, selling 55 shirts for $605.

The domain the situation allows

A model is a description of something real, and real quantities have limits. The height function above only means anything for 0 <= t <= 5: before the throw there is no flight, and after the ball lands the formula keeps producing numbers that describe nothing.

A fencing problem is the same. With 60 feet of fence along three sides of a rectangle — a barn wall supplying the fourth — the area is A(w)=w(60-2w). Widths of 0 or 30 give no pen at all, and anything outside that range gives a negative dimension.

Common mistake — Answering when the question asked how much

\"A ball's height is h(t)=-16t^2+64t+80. What is its maximum height?\" A student computes t=-(64)/(2(-16))=2 and writes 2 feet.

The algebra was flawless. The reporting was not. The formula -(b)/(2a) produces an input — here a time. The maximum value is what comes out when you put that input back in: h(2)=144 feet.

The error runs the other way too. Asked when the ball is highest, students who have learned to substitute sometimes answer 144 seconds, which is nearly four minutes in the air.

How to catch it: Write the unit next to your answer before you write the answer down. The axis formula produced seconds; the question asked for feet. If the unit you have is not the unit the question wants, you are exactly one substitution away from being finished.