SAT-36: Finding Maximum and Minimum Values of a Quadratic

SAT-36: Finding Maximum and Minimum Values of a Quadratic

Description: Many SAT word problems ask for a maximum profit, maximum height, or minimum cost. All of these are the vertex of a parabola. This lesson shows how to find it from any form.

The key idea

The max or min value of a quadratic happens at the vertex. The vertex's x = h tells you where; the y = k tells you the max/min value itself.

(Oʻzbekcha: kvadrat funksiyaning eng katta yoki eng kichik qiymati doim uchida (vertex) boʻladi.)

Method 1 — from vertex form

If y = a(x − h)² + k, the max/min value is simply k (review SAT-35).

Method 2 — from standard form

For y = ax² + bx + c, the vertex's x-coordinate is:

x = −b / (2a)

Plug that x back in to get the max/min value (the y). (Oʻzbekcha: avval x = −b/2a ni top, keyin uni qoʻyib y ni hisobla.)

Worked Example

A ball's height is h(t) = −16t² + 64t. What is its maximum height?

• a = −16, b = 64. Time of max: t = −b/(2a) = −64 / (2·−16) = −64 / −32 = 2 seconds.
• Max height: h(2) = −16(2)² + 64(2) = −64 + 128 = 64.

Tip: a < 0 means it opens down, so the vertex is a maximum — exactly what a "maximum height" problem needs.

(Oʻzbekcha: "maksimal balandlik" masalalarida a manfiy boʻlganligi uchun uch eng katta qiymatni beradi.)

Practice

Find the minimum value of y = x² − 6x + 5.

Key words — Kalit soʻzlar

• Maximum — eng katta qiymat
• Minimum — eng kichik qiymat
• Vertex — uch (choʻqqi)
• Profit — foyda
• Cost — xarajat
• Height — balandlik
• Standard form — standart koʻrinish
• Substitute / Plug in — (qiymatni) qoʻyish
• Coordinate — koordinata

Summary

• Max/min of a quadratic = the vertex.
• Vertex form: the value is k directly.
• Standard form: find x = −b/(2a), then plug in to get the value.
• a > 0 → minimum; a < 0 → maximum.