Quadratic Modeling on the SAT
Maximums, Minimums, and Real Stories
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Quadratic Modeling
Quadratic models show up whenever something rises and then falls, like a ball thrown into the air, or profit that grows and then declines. The SAT uses these scenarios because the vertex has a clear real world meaning.
In this lesson, you will learn how to interpret the vertex as a maximum or minimum, translate the model into a question about time or height, and use Desmos when the arithmetic is messy. We will also practice reading which value the question actually wants.
A Simple Definition Unlocks Quadratic Modeling
A quadratic model looks like $h(t) = at^2 + bt + c$. The vertex gives the highest or lowest value depending on whether $a$ is negative or positive.
Always connect the math back to the context. If $t$ represents time, the vertex $t$ value is when the maximum occurs, and the vertex $h$ value is the maximum height. Mixing those two is a very common error.
Work Through Quadratic Modeling Step by Step
For $h(t) = -16t^2 + 32t + 5$, when is the maximum height?
Find the maximum height for $h(t) = -16t^2 + 32t + 5$.
Identify the coefficient $a$ from the quadratic.
Identify the coefficient $b$ from the quadratic.
Use the vertex formula to find the time of the maximum or minimum.
Substitute the values into the formula.
Simplify the expression to make the next step clear.
Plug in to find height
Compute the maximum height by evaluating the function at the vertex time.
Use Desmos to Check Quadratic Modeling
For $h(t) = -16t^2 + 32t + 5$, when is the maximum height?
Desmos is excellent for quadratic models. Graph the function and click the vertex to read the maximum or minimum directly.
h(t) = -16t^2 + 32t + 5
Algebra is best when the question requires an exact expression. Desmos is faster when you need a quick numeric value or want to confirm your work.
Expert move: Graph the quadratic and click the $x$-intercepts and vertex to read solutions and key features; the graph makes it clear when there are 0, 1, or 2 real roots.
Precision check: Use Desmos for decimal answers or verification, but convert to a fraction if the choices are exact and apply any context restrictions.
- Desmos features used: graphing, vertex display.
- Common mistake: mixing up the time of the maximum with the maximum height.
Practice Quadratic Modeling with SAT-Style Questions
Interpret the model and read the vertex carefully.
The function models height. When is the maximum height reached?
A ball is modeled by . What is the maximum height?
A revenue model is . At what is revenue maximized?
For , what is the height at the vertex?
Key Takeaways to Remember for Quadratic Modeling
- The vertex gives the maximum or minimum in real contexts.
- Find the vertex time with $-\frac{b}{2a}$, then plug in.
- Desmos is a fast way to read maximums and minimums.
