Skill 12 of 12

Model and solve problems with quadratics

Write and solve quadratic equations for real situations (projectile motion, area, revenue) and interpret the vertex and roots in context.

Common Core standards: A-CED.1F-IF.4
Builds on: Key features of quadratic functions and graphs
Solve quadratics with the quadratic formula

Common mistakes

What students most often get wrong with this skill:

Reporting a negative time or length as a valid answer
Confusing the time of maximum height with the maximum height
Setting up the area equation with the wrong expression for one side

A worked example

From the lesson, with every step explained.

Example 1: Evaluate a model

Problem. A ball is thrown upward. Its height in feet after $t$ seconds is

h(t) = -16t^2 + 48t + 4

Find the height at $t = 2$ .

Plan. "Height at $t=2$ " means substitute $2$ for $t$ and compute. Follow the order of operations: power first, then multiply, then add.

1) Substitute.

h(2) = -16(2)^2 + 48(2) + 4

2) Square first, because powers come before multiplying.

h(2) = -16(4) + 48(2) + 4

3) Multiply.

h(2) = -64 + 96 + 4

4) Add.

h(2) = 36

Answer. After 2 seconds the ball is 36 feet high.

Check. Recompute in a different order: $-64 + 96 = 32$ , and $32 + 4 = 36$ . It matches. A height of 36 ft is also sensible, since the ball started at 4 ft and went up.

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