Vertex of a Parabola

Sommaire

In short: Vertex of a parabola refers to its single turning point, the lowest spot on a bowl or the highest on an arch, located at x = −b/(2a).

Algebra 1Functions and GraphsF-IF.C.8a

What is Vertex of a Parabola?

A U-shaped graph turns around exactly once, and that spot is the vertex of a parabola. It is where a profit curve peaks, where a tossed ball reaches its top, and where a valley floor sits.

For $y = ax^2 + bx + c$ the x-coordinate is $x = -\frac{b}{2a}$, and substituting it back produces the height. The vertex sits on the mirror line, and its height is the maximum when $a$ is negative or the minimum when $a$ is positive.

How to spot it

The wording asks for the vertex of a parabola, disguised as a greatest height, a best profit or a least cost.
A graph shows a lowest or highest point and the coordinates are wanted.
A quadratic in the form $ax^2 + bx + c$ arrives with no picture, yet the turning point is needed.
Two twin points share a height, so the midpoint between their x-values gives the turning x.

The method

Pick a and b out of $y = ax^2 + bx + c$.
Calculate $x = -\frac{b}{2a}$.
Put that x into the equation to obtain y.
Write the pair $(x, y)$ and call it a maximum for negative a or a minimum for positive a.

For y = x² − 6x + 11 the lowest point is the vertex (3, 2).

Vertex of a Parabola: solved example

Problem

A lemonade stand’s weekly profit is $P(x) = -2x^2 + 40x – 110$ dollars when a cup sells for x dollars. Which price gives the largest profit, and how large is it?

Solution

The curve opens down because $a = -2$, so the turning point is a peak. The price is $x = -\frac{40}{2(-2)} = 10$.

Then $P(10) = -200 + 400 – 110 = 90$. The peak is $(10, 90)$: selling cups at $10 earns the best profit, $90.

Try it

Question 1

Locate the turning point of $y = x^2 + 4x + 9$.

Show the answer

$x = -\frac{4}{2} = -2$ and $y = 4 – 8 + 9 = 5$, so $(-2, 5)$.

Question 2

How high can $y = -x^2 + 10x$ go?

Show the answer

The peak is at $x = 5$, where $y = -25 + 50 = 25$.

Vertex of a Parabola vs. Axis of Symmetry

The turning point is a spot with two coordinates, like $(3, 2)$. The mirror line is a vertical line through that spot, written $x = 3$. Both share the x-value, yet one is a location and the other is a line that folds the curve in half.

Axis of Symmetry

Keep going

Lessons

Graphing Quadratic Functions and Transformations

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Related words

Whole dictionary: A–Z · Algebra 1 · Functions and Graphs

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