Connect equation forms to graph features (factored/standard/vertex form for quadratics). | Nonlinear functions | SAT Math

Matching the three quadratic forms to the graph features each one reveals most directly.

核心知识

Standard form $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐$ shows the y-intercept ( $𝑐$ ) and opens up/down (sign of $𝑎$ ). Vertex form $𝑎 (𝑥 - ℎ) 2 + 𝑘$ shows the vertex $(ℎ, 𝑘)$ . Factored form $𝑎 (𝑥 - 𝑝) (𝑥 - 𝑞)$ shows the x-intercepts ( $𝑝$ and $𝑞$ ).

深入理解

The SAT will give you a quadratic in one particular form and ask about a feature that form reveals — or ask you to convert to the form that makes the answer obvious.

Standard form $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐$ :

$𝑐$ is the y-intercept.
The sign of $𝑎$ tells you direction: $𝑎 > 0$ opens up, $𝑎 < 0$ opens down.
You can find the vertex using $𝑥 = - 𝑏 2 𝑎$ .

Vertex form $𝑎 (𝑥 - ℎ) 2 + 𝑘$ :

$(ℎ, 𝑘)$ is the vertex — the maximum or minimum point.
Best for questions about the peak, lowest point, or axis of symmetry ( $𝑥 = ℎ$ ).

Factored form $𝑎 (𝑥 - 𝑝) (𝑥 - 𝑞)$ :

$𝑝$ and $𝑞$ are the x-intercepts (roots/zeros).
The axis of symmetry is at $𝑥 = 𝑝 + 𝑞 2$ , midway between the roots.

Knowing which form to use — or when to convert — saves significant time.

常见误解

Trying to read x-intercepts from standard form without factoring or using the quadratic formula.
Confusing the vertex with the y-intercept — they are the same only when the axis of symmetry passes through $𝑥 = 0$ .
Forgetting that in factored form $𝑎 (𝑥 - 𝑝) (𝑥 - 𝑞)$ , the roots are $𝑝$ and $𝑞$ (not $- 𝑝$ and $- 𝑞$ ).

题目

示例解析

The function $𝑓 (𝑥) = 2 (𝑥 - 3) (𝑥 + 1)$ models a certain relationship. What are the x-intercepts of the graph of $𝑦 = 𝑓 (𝑥)$ ?

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