Creating and Using Quadratic Functions in Context
A quadratic function models anything with constant acceleration or two multiplied lengths, like falling objects or fenced areas. Its vertex gives the maximum or minimum, and its zeros give the start/end times or widths.
A ball in the air speeds up as it falls, because gravity constantly pulls it down. That changing speed is exactly what makes height a quadratic function of time rather than a straight line: the -16t2 term (half of gravity's pull, in feet) curves the graph into a parabola, while the other terms set the launch speed and starting height. Anything with that kind of constant push or pull, or anything built from multiplying two lengths that trade off against each other, like the sides of a fenced rectangle, turns into a quadratic in the same way.
Turning the story into a function means naming one quantity x and writing everything else in terms of it. For fencing problems, if you have a fixed total length of fencing and one side is shared with a wall, the two remaining sides split what's left, so each one becomes (P - x)/2. Multiply that by x and you get area as a function of x, a downward parabola. For falling objects, h(t) = -16t2 + vt + h0 comes from the initial velocity v and starting height h0 you're given. Once you have the function, the vertex -b/(2a) answers every 'best' or 'worst' question: maximum area, maximum height, minimum cost. For timing or distance, you instead set the function equal to zero and solve.
This topic connects directly to the next skill of reading a quadratic's vertex form and zeros algebraically: the fencing and projectile problems are just quadratics wearing a story, and the same vertex formula and factoring tools apply to both. Recognizing which parts of a problem play the role of a, b, and c is the whole translation step.
Key terms
- Vertex
- The highest or lowest point of a parabola, found at t = -b/(2a) for a quadratic in standard form.
- Zero
- A value of the input where the function's output equals 0, such as when height hits the ground.
- Standard form
- The form ax2 + bx + c, where a, b, and c are numbers set by the problem's story.
- Parabola
- The U-shaped or dome-shaped curve traced by any quadratic function.
Watch it explained
- How it worksHow do you turn a story about fenced area or falling height into an actual quadratic function?
- How it worksHow does the vertex of a quadratic model become the answer to every maximum or minimum question?
- Worked exampleFind when a ball with height h(t) = -16t2 + 64t + 5 finally hits the ground.
- Worked exampleFind the largest rectangular area you can fence off with 80 feet of fencing against a wall.
- Common mistakeWhy doesn't a projectile reach its highest point at the same moment it lands?
Check yourself
End of topic.