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The math of every arch

Quadratic functions, visually

A quadratic is the rule behind every arch: a thrown ball, a bridge, a jump. Read its top from the vertex form and its landing spots from the factored form. Drag the sliders and watch Pip stand on the vertex.

Free to start. Coming soon to the App Store.

y = 1(x − 2)² + −1
  1. vertex (2, −1)
  2. axis of symmetry x = 2
  3. lands at x = 1 and x = 3
It opens up like a bowl, so the vertex is the lowest point. It lands on the x-axis twice.
Idea 1 · Bowl or arch, wide or narrow

Bowl or arch, wide or narrow

The number in front, a, sets the shape of y = ax². Positive opens up like a bowl. Negative opens down like an arch. A minus does not move the shape down: it flips it.

A bigger number makes the parabola narrower, and a number between 0 and 1 makes it wider. A parabola also grows faster than a line: x² is 1, 4, 9, 16, while 2x is only 2, 4, 6, 8.

Aha!

The sign in front says bowl or arch, and the size says wide or narrow.

Idea 2 · The top and the mirror line

The top and the mirror line

The vertex is the top of an arch or the bottom of a bowl. In y = a(x − h)² + k it is the point (h, k), and the sign of h is the one people get wrong.

The vertical line x = h through it is the axis of symmetry: every point on one side has a mirror twin on the other, at the same height.

Aha!

The vertex form shows the top at (h, k), and every point has a mirror twin across the line x = h.

Idea 3 · Where it lands

Where it lands: roots from the factored form

The x-intercepts, also called roots or zeros, are the landing spots where y = 0. In the factored form y = (x − 1)(x − 5) they are x = 1 and x = 5.

Mind the signs: a bracket (x + 2) gives the root −2. A product is zero only when one bracket is zero. This is the zero product property, and it works when the other side is 0.

Aha!

A product is zero only if one bracket is zero, so the landing spots are the numbers that make a bracket zero.

Idea 4 · How many landings?

How many landings? 0, 1 or 2

A line can cross a parabola at 0, 1 or 2 points, and the x-axis is just one such line. A quadratic can land twice, touch down once, or never land at all.

The discriminant b² − 4ac in the quadratic formula tells you which: positive gives two roots, zero gives one, and negative gives none.

Aha!

A parabola and a line can meet twice, once or never, so expect 0, 1 or 2 answers, not always exactly one.

Quick answers

Questions about quadratics

What is the vertex of a parabola?

The turning point: the lowest point of a bowl or the highest point of an arch. In vertex form y = a(x − h)² + k it is (h, k). In y = ax² + bx + c the x-coordinate is −b ÷ 2a.

How do you find the roots of a quadratic?

Set y = 0 and solve. Factor and use the zero product property, or use the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. On a graph, the roots are where the parabola crosses the x-axis.

What does the discriminant tell you?

The number b² − 4ac. If it is positive the parabola crosses the x-axis twice, if it is zero it touches the axis once, and if it is negative it never reaches the axis, so there are no real roots.

How can you tell if a parabola opens up or down?

Look at the number in front of x². Positive opens up, like a bowl with a lowest point. Negative opens down, like an arch with a highest point.

Now play it

Try it in Pip: Math Adventure

World 5 of Pip: Math Adventure is all about quadratics. You launch Pip over walls, and the jump he takes is the parabola you chose.

It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.

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