Model real-world situations with quadratic functions and solve, interpreting the vertex as a maximum or minimum and the zeros as start or end points, and rejecting solutions that do not fit the context (MA.912.AR.3.6, MA.912.AR.3.9).

What this topic is asking

MA.912.AR.3 asks you to model and solve real situations with quadratics, then interpret. The two staples are projectile motion (height over time) and area (length times width). The skill the B.E.S.T. Algebra 1 EOC rewards is deciding when to use the vertex (a maximum or minimum) versus the zeros (a start or end), and rejecting solutions that cannot happen.

Projectile motion

A falling or thrown object's height follows a downward parabola. The standard model (feet and seconds) is:

where $v_0$ is the initial upward velocity and $h_0$ the initial height. This model is not on the reference sheet, so know its shape. The peak height is at the vertex ($t = \frac{-b}{2a}$), and the object lands when $h(t) = 0$ (a zero).

Area models

For area problems, translate the relationship into length and width. "A rectangle is 5 longer than wide with area 36" becomes $w(w + 5) = 36$, then $w^2 + 5w - 36 = 0$, solved by factoring. A dimension can never be negative, so reject the negative root.

Vertex versus zeros: which to use

The wording points to the right feature:

• "Maximum," "minimum," "greatest," "how high," "most profit" -> the vertex (report the $y$-value).
• "When does it land," "hits the ground," "breaks even," "runs out" -> a zero (where the output is $0$).

How the B.E.S.T. EOC examines this topic

• Equation editor and number entry. Compute a maximum height, a landing time, or a dimension.
• Multiple choice. Identify the maximum, a zero, or the correct realistic solution.
• Context items. Set up the model and interpret, with units.

A clarifying idea: a quadratic application is a parabola dressed as a story. The vertex is the turning point of the story (the most or least), and the zeros are where the quantity reaches zero (launch and landing). Matching the question's wording to the vertex or the zeros is the whole interpretive task.

Why you reject some solutions

A quadratic almost always has two algebraic solutions, but the context frequently makes one impossible, and discarding it is part of the credit, not an afterthought. A negative time would be before the object was launched, a negative length or width has no physical meaning, and a negative count of people or items cannot occur. The algebra does not know the context, so it returns both roots; your job is to apply the real-world constraint and keep only the root that fits. This is why a landing-time problem keeps the positive root and an area problem keeps the positive dimension. Stating the kept answer with units ("3.1 seconds," "5 meters") confirms you have interpreted, not just solved, which is exactly what the B.E.S.T. modeling items reward.

Try this

Q1. A ball's height is $h(t) = -16t^2 + 64t$. When does it land? [2 points]

• Cue. $-16t(t - 4) = 0$, so $t = 0$ or $t = 4$; it lands at $t = 4$ s.

Q2. A rectangle is 2 m longer than wide with area 24. Find the width. [2 points]

• Cue. $w(w + 2) = 24 \Rightarrow (w + 6)(w - 4) = 0$; width $4$ m.