Parabola

Accueil » Parabola
Parabola: meaning, method and solved example (Algebra 1) – Maths PDF

In short: A parabola is the symmetric bowl or arch you get by graphing a quadratic, bending away from one turning point in a single direction.

Algebra 1Functions and GraphsF-IF.C.7a

What is Parabola?

Trace the path of a tossed ball or the arc of water from a drinking fountain and you are following a parabola. It is the picture of \(y = ax^2 + bx + c\), and it has a mirror line, so the left half copies the right half.

Four features matter. The vertex is the turning point, a maximum if the curve opens down and a minimum if it opens up. A vertical mirror line passes through that vertex. The x-intercepts mark where the curve meets the horizontal axis, and the y-intercept equals \(c\).

How to spot it

  • A graph resembles a bowl or an arch with exactly one place where it turns around.
  • A quadratic equation must be graphed, or its greatest or least value found.
  • Something flies on a curved path under gravity, or a surface looks like a satellite dish.
  • Points on the curve come in twin pairs sitting at the same height.

The method

  1. Check the sign of a to see whether the parabola opens up or down.
  2. Use \(x = -\frac{b}{2a}\) for the mirror line, then put that x into the equation for the vertex height.
  3. Compute the y-intercept (set x to 0) and the x-intercepts (set y to 0).
  4. Plot those points with their twins, then draw one smooth curve through them.
vertex (3, 4)(1, 0)(5, 0)axis of symmetry x = 3y = −x² + 6x − 5a = −1: opens down
This parabola opens down, with vertex (3, 4) and x-intercepts at 1 and 5.

Parabola: solved example

Problem

Take \(y = -x^2 + 6x – 5\) and find which way it opens, its mirror line, its turning point and its intercepts.

Solution

Since \(a = -1\), the curve opens down. The mirror line is \(x = -\frac{6}{2(-1)} = 3\), and \(y = -9 + 18 – 5 = 4\), so the turning point \((3, 4)\) is a maximum.

At \(x = 0\) the height is \(-5\). Solving \(-x^2 + 6x – 5 = 0\) gives \((x – 1)(x – 5) = 0\), so the curve meets the horizontal axis at 1 and 5.

Try it

Question 1

Locate the turning point of \(y = 3x^2 – 12x\).

Show the answer

\(x = -\frac{-12}{6} = 2\) and \(y = 12 – 24 = -12\), so \((2, -12)\).

Question 2

Does the graph of \(y = x^3\) count as one?

Show the answer

No. The top power is 3 and there is no mirror line, so it is a different curve.

Parabola vs. Quadratic Function

The function is the rule containing an x-squared term; the curve is the picture that rule draws. Tables and equations belong to the rule. The opening direction, the turning point and the mirror line describe the shape on the graph.

Quadratic Function

Keep going

Lessons

Worksheets and quizzes

Related words

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

Voter.. page