SAT quadratics: roots, vertices, and what the graph means

By Graph Math Practice editorial team · Reviewed 2026-09-05

Roots and vertices answer different questions

A root is an x-value that makes the function equal zero. On a graph, it lies where the curve meets the horizontal axis. A vertex is the turning point of a parabola. Its x-coordinate tells you where the turn occurs; its y-coordinate tells you the minimum or maximum output.

Consider y = (x - 3)² - 4. The vertex is (3, -4). The roots are 1 and 5. If a question asks for the minimum value of the function, -4 is correct. If it asks for the greater zero, 5 is correct. A quick sketch showing all three features can prevent a coordinate mix-up.

Use vertex form to predict the picture

In y = a(x - h)² + k, the vertex is (h, k). When a is positive, the parabola opens upward and k is a minimum. When a is negative, the curve opens downward and k is a maximum. Notice the subtraction inside the parentheses: x + 2 can be written as x - (-2), so its horizontal vertex coordinate is -2.

For y = -2(x - 4)² + 18, the squared expression is always nonnegative. Multiplying it by -2 produces a value no greater than zero. Adding 18 means y cannot exceed 18. The maximum occurs at x = 4. You can establish this without graphing or expanding the expression.

Check the roots exactly

Setting (x - 3)² - 4 equal to zero gives (x - 3)² = 4. Both 2 and -2 square to 4, so x = 5 or x = 1. Keeping both possibilities matters when a question asks for the smaller root, the greater root, or their sum.

A displayed graph may round a coordinate. Substitute the proposed answer into the original expression to check it, especially when options are close together. If the question provides an exact expression, avoid unnecessarily rounding intermediate steps. Our graph workspace reports approximate numerical values and is not a replacement for exact algebra.

Connect the output to the story

In a model of height over time, the y-coordinate may represent height while x represents elapsed time. The maximum height and the time of maximum height are therefore different answers with different units. Label those units before entering anything into a calculator.

Use the quadratic drills to separate the two decisions: first identify the requested feature, then calculate it. If a root is negative but the model describes time after launch, the context may exclude that negative value. A mathematical solution and an admissible real-world answer are not always the same thing.

Put the idea into practice.

Try eight original questions with explanations after each answer.

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References and further practice

References establish the test context. The examples and teaching explanations here are original. See our methodology for coverage and limitations.