Determine key features (e.g., y-intercept, vertex) from a nonlinear graph or equation. | Nonlinear functions | SAT Math

Extracting intercepts, vertex, and other features algebraically from equations.

核心知识

The y-intercept is always $𝑓 (0)$ . For quadratics, the vertex is at $𝑥 = - 𝑏 2 𝑎$ in standard form or read directly as $(ℎ, 𝑘)$ in vertex form $𝑎 (𝑥 - ℎ) 2 + 𝑘$ . The x-intercepts come from setting $𝑓 (𝑥) = 0$ .

深入理解

When you don't have a graph, you need to pull features straight from the equation.

y-intercept: Plug in $𝑥 = 0$ . In $𝑓 (𝑥) = 2 𝑥 2 - 8 𝑥 + 6$ , the y-intercept is $𝑓 (0) = 6$ . In $𝑔 (𝑥) = 5 \cdot 3 𝑥$ , the y-intercept is $𝑔 (0) = 5$ .

Vertex of a quadratic in standard form $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐$ : the x-coordinate is $- 𝑏 2 𝑎$ , then plug back in for the y-coordinate. In vertex form $𝑎 (𝑥 - ℎ) 2 + 𝑘$ , the vertex is $(ℎ, 𝑘)$ directly — no calculation needed.

x-intercepts: Set the equation equal to zero and solve. Factoring is fastest when possible; otherwise use the quadratic formula. For exponentials like $𝑎 \cdot 𝑏 𝑥$ (with $𝑎 \neq 0$ ), there are no x-intercepts because $𝑏 𝑥$ is never zero.

分步讲解

To find the y-intercept: substitute $𝑥 = 0$ into the equation.
To find the vertex (quadratic in standard form): compute $𝑥 = - 𝑏 2 𝑎$ , then evaluate $𝑓 (- 𝑏 2 𝑎)$ .
To find x-intercepts: set $𝑓 (𝑥) = 0$ and solve by factoring, completing the square, or the quadratic formula.

常见误解

Forgetting the negative sign in $- 𝑏 2 𝑎$ — this is the most common arithmetic slip on vertex problems.
In vertex form $𝑎 (𝑥 - ℎ) 2 + 𝑘$ , reading the sign of $ℎ$ incorrectly: $(𝑥 - 3) 2$ means $ℎ = 3$ , not $ℎ = - 3$ .
Assuming exponential functions have x-intercepts. They approach zero but never reach it (unless shifted vertically).

题目

示例解析

What is the vertex of the parabola defined by $𝑓 (𝑥) = - 3 (𝑥 + 4) 2 + 7$ ?

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