Identify and interpret key graph features (intercepts, vertex/extrema, maximum/minimum). | Nonlinear functions | SAT Math

Reading intercepts, vertex, and extrema from graphs and explaining what they mean.

核心知识

Every key graph feature answers a specific question: x-intercepts → where does the output equal zero? y-intercept → what is the starting value? Vertex/extrema → what is the highest or lowest value the function reaches, and at what input?

深入理解

SAT graph-reading questions show you a curve and ask what a particular feature means in the given context.

For x-intercepts, set $𝑦 = 0$ . On a height-vs-time graph, these are the moments when the object is at ground level. On a profit-vs-price graph, these are the break-even prices.

The y-intercept is the output when $𝑥 = 0$ . This is almost always the initial or baseline value.

The vertex of a parabola (or the peak/valley of any curve) gives the extreme value. If the curve opens downward, the vertex is the maximum. A question might ask: "At what time does the ball reach its greatest height?" — that's the x-coordinate of the vertex. "What is the greatest height?" — that's the y-coordinate.

When interpreting, always match the feature back to the units on each axis. The x-axis label tells you what the x-coordinate means; the y-axis label tells you what the y-coordinate means.

常见误解

Reading the x-coordinate of the vertex when the question asks for the maximum value (which is the y-coordinate), or vice versa.
Confusing x-intercepts with the vertex — they answer different questions.
Assuming the y-intercept is always zero; it's whatever value the function takes at $𝑥 = 0$ .

题目

示例解析

The graph of $𝑦 = 𝑓 (𝑥)$ models the profit, in thousands of dollars, a company earns when it sets its product price at $𝑥$ dollars. The graph is a downward-opening parabola with vertex at $(25, 40)$ and x-intercepts at $𝑥 = 5$ and $𝑥 = 45$ . What is the maximum profit the company can earn?

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