Previously, students used simple quadratic functions to describe how an object falls over time given the effect of gravity. In this lesson, they build on that understanding and construct quadratic functions to represent projectile motions. Along the way, they learn about the zeros of a function and the vertex of a graph. They also begin to consider appropriate domains for a function given the situation it represents.
Students use a linear model to describe the height of an object that is launched directly upward at a constant speed. Because of the influence of gravity, however, the object will not continue to travel at a constant rate (eventually it will stop going higher and will start falling), so the model will have to be adjusted (MP4). They notice that this phenomenon can be represented with a quadratic function, and that adding a squared term to the linear term seems to “bend” the graph and change its direction.
- Create graphs of quadratic functions that represent a physical phenomenon and determine an appropriate domain when graphing.
- Identify and interpret (orally and in writing) the meaning of the vertex of a graph and the zeros of a function represented in tables and graphs.
- Write and interpret (orally and in writing) quadratic functions that represent a physical phenomenon.
- Let’s look at the objects being launched in the air.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can create quadratic functions and graphs that represent a situation.
- I can relate the vertex of a graph and the zeros of a function to a situation.
- I know that the domain of a function can depend on the situation it represents.
A function where the output is given by a quadratic expression in the input.
vertex (of a graph)
The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing or vice versa. It is the highest or lowest point on the graph.
zero (of a function)
A zero of a function is an input that yields an output of zero. If other words, if \(f(a) = 0\) then \(a\) is a zero of \(f\).