Graph y = (x - 3)^2 + 1 from y = x^2 Step by Step

Step by step

The question

Question: graph y = (x − 3)² + 1 from y = x².

y = (x − 3)² + 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

Inside, next to x, we have − 3.

x = 3 makes the inside 0.

So the turning point moves to x = 3: right 3.

x − 3 inside → right 3

Careful!

Inside, it goes the opposite way to the sign.

Careful: − 3 inside goes right, not left

Outside, at the end, + 1 adds 1 to every y.

So the graph moves up 1.

+ 1 outside → up 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 right 3.

Then up 1.

The new turning point is (3, 1).

4 Draw the new graph

The shape stays the same.

From the turning point, 1 right, 1 up.

And 1 left, 1 up.

Join them with the same U.

This is y = (x − 3)² + 1.

5 Check

Let's check.

Put x = 3 in the new rule.

We get y = 1: the turning point (3, 1).

x = 3: y = (3 − 3)² + 1 = (0)² + 1 = 0 + 1

= 1

And x = 4 gives y = 2: the point (4, 2) we drew.

x = 4: y = (4 − 3)² + 1 = (1)² + 1 = 1 + 1

= 2

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