Vertex Form on the SAT

Why the SAT Emphasizes Vertex Form

Vertex form is the fastest way to read the most important feature of a parabola, its turning point. The SAT often asks for the maximum, minimum, or the axis of symmetry, all of which are tied to the vertex.

In this lesson, you will learn how to rewrite a quadratic into vertex form by completing the square, and how to read the vertex correctly. We will also address the common confusion about the sign inside $(x - h)$.

A Simple Definition Unlocks Vertex Form

Vertex form is $y = a(x - h)^2 + k$. The vertex is $(h, k)$, and the axis of symmetry is $x = h$. The number $a$ tells you whether the parabola opens up or down.

Completing the square is the algebra tool that converts standard form into vertex form. Once you do that, you can read the vertex instantly instead of doing extra calculation.

Work Through Vertex Form Step by Step

Convert $y = x^2 - 6x + 5$ to vertex form.

Use Desmos to Check Vertex Form

Desmos shows the vertex automatically when you click the parabola. This is very fast when a question asks for the vertex or maximum value.

Algebra is required when the question asks for the exact vertex form. Desmos is faster for reading the vertex coordinates quickly.

Expert move: Graph the quadratic and click the $x$-intercepts and vertex to read solutions and key features; the graph makes it clear when there are 0, 1, or 2 real roots.

Precision check: Use Desmos for decimal answers or verification, but convert to a fraction if the choices are exact and apply any context restrictions.

• Desmos features used: graphing, vertex display.
• Common mistake: using $(-h, k)$ instead of $(h, k)$ for the vertex.

Practice Vertex Form with SAT-Style Questions

Convert or interpret vertex form in each question.

Key Takeaways to Remember for Vertex Form

• Vertex form makes the turning point visible as $(h, k)$.
• Complete the square to convert from standard form.
• Desmos shows the vertex quickly for graph interpretation.