Graph y = √(x - 1) - 3 from y = √x Step by Step

Step by step

The question

Question: graph y = √(x − 1) − 3 from y = √x.

y = √x − 1 − 3

We know the graph of y = √x.

It is half a U, lying on its side.

we know: y = √x

1 Read the changes

Inside, next to x, we have − 1.

x = 1 makes the inside 0.

So the start point moves to x = 1: right 1.

x − 1 inside → right 1

Outside, at the end, − 3 takes 3 away from every y.

So the graph moves down 3.

− 3 outside → down 3

Careful!

Inside, it goes the opposite way to the sign.

Careful: − 1 inside goes right, not left

2 Start from y = √x

Draw the grid.

Draw y = √x first.

It goes through (0, 0), (1, 1) and (4, 2).

3 Move the points

The start point (0, 0) moves right 1.

Then down 3.

(0, 0) goes to (1, −3).

(1, 1) goes to (2, −2).

(4, 2) goes to (5, −1).

4 Draw the new graph

Join the new points with the same shape.

This is y = √(x − 1) − 3.

5 Check

Let's check.

Put x = 1 in the new rule.

We get y = −3: the point (1, −3) we drew.

x = 1: y = √1 − 1 − 3 = √0 − 3 = 0 − 3 = −3

And x = 5 gives y = −1: the point (5, −1).

x = 5: y = √5 − 1 − 3 = √4 − 3 = 2 − 3 = −1

Let's look back.

First: read the changes.

Then: start from y = √x.

Then: move the points.

Then: draw the new graph.

Last: check.

Answer