Graph y = 2x^2 - 1 from y = x^2 Step by Step

Step by step

The question

Question: graph y = 2x² − 1 from y = x².

y = 2x² − 1

We know the graph of y = x².

It is a U, with its turning point at (0, 0).

we know: y = x²

1 Read the changes

The 2 in front makes every y 2 times as big.

The U gets narrower.

2 in front → every y × 2

Outside, at the end, − 1 takes 1 away from every y.

So the graph moves down 1.

− 1 outside → down 1

2 Start from y = x²

Draw the grid.

The x-axis goes across.

The y-axis goes up and down.

They meet at 0.

That point is called the origin.

Draw y = x² first.

Its turning point is at (0, 0).

3 Move the turning point

Move it down 1.

The new turning point is (0, −1).

4 Draw the new graph

Every y is 2 times as far.

From the turning point, 1 right, 2 up.

And 1 left, 2 up.

Join them with a narrower U.

This is y = 2x² − 1.

5 Check

Let's check.

Put x = 0 in the new rule.

We get y = −1: the turning point (0, −1).

x = 0: y = 20² − 1 = 2 × 0 − 1 = 0 − 1 = −1

And x = 1 gives y = 1: the point (1, 1) we drew.

x = 1: y = 21² − 1 = 2 × 1 − 1 = 2 − 1 = 1

Let's look back.

First: read the changes.

Then: start from y = x².

Then: move the turning point.

Then: draw the new graph.

Last: check.

Answer