Build or use a quadratic model; interpret vertex and intercepts in context. | Nonlinear functions | SAT Math

Setting up quadratic equations from word problems and reading meaning from their vertex and intercepts.

核心知识

In a quadratic model $𝑓 (𝑥) = 𝑎 (𝑥 - ℎ) 2 + 𝑘$ , the vertex $(ℎ, 𝑘)$ gives the maximum or minimum value and when it occurs. The x-intercepts (zeros) mark where the output equals zero — often a start/end time or break-even point.

深入理解

SAT quadratic-modeling questions usually describe a physical or business scenario and ask you to find or interpret a specific feature.

The vertex answers "what is the greatest (or least) value, and at what input does it happen?" If $𝑎 < 0$ , the parabola opens down and the vertex is a maximum — common for projectile height or profit. If $𝑎 > 0$ , it opens up and the vertex is a minimum — common for cost or distance.

The x-intercepts answer "when does the quantity equal zero?" For a ball thrown upward, the intercepts are launch time and landing time.

The y-intercept (set $𝑥 = 0$ ) gives the starting value — initial height, initial cost, etc.

When building a model from scratch, identify which variable is independent (usually time or quantity), write revenue as $p r i c e \times q u a n t i t y$ or height as $- 16 𝑡 2 + 𝑣 0 𝑡 + ℎ 0$ , and simplify.

分步讲解

Identify the independent variable (time, quantity, etc.) and the dependent variable (height, revenue, etc.).
Write the relationship using the given information — often a product of two linear expressions or a standard projectile formula.
Simplify to standard or vertex form as needed.
Read the requested feature: vertex for the maximum or minimum value, intercepts for zeros, y-intercept for initial value.

常见误解

Mixing up x-coordinate and y-coordinate of the vertex — the x-coordinate is when the maximum/minimum happens; the y-coordinate is the max/min value.
Forgetting that a < 0 means maximum and a > 0 means minimum.
Using -16t^2 (feet) when the problem uses meters, or vice versa.

题目

示例解析

A ball is launched upward from a platform. Its height $ℎ$ , in feet, after $𝑡$ seconds is modeled by $ℎ (𝑡) = - 16 𝑡 2 + 64 𝑡 + 80$ . What is the maximum height the ball reaches?

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