Steply

Graph transformations

Algebra 2 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Read the changes
  2. Start from y = x²
  3. Move the turning point
  4. Draw the new graph
  5. Check

The steps for Graph y = x^2 + 3 from y = x^2. Every solution names its steps, so you always know what comes next.

Worked examples

Graph y = x^2 + 3 from y = x^2 Easy

The question

Question: graph y = x² + 3 from y = x².

y = x2 + 3

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

Outside, at the end, + 3 adds 3 to every y.

So the graph moves up 3.

+ 3 outside → up 3

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 up 3.

The new turning point is (0, 3).

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 = x² + 3.

5 Check

Let's check.

Put x = 0 in the new rule.

We get y = 3: the turning point (0, 3).

x = 0: y = 02 + 3 = 0 + 3 = 3

And x = 1 gives y = 4: the point (1, 4) we drew.

x = 1: y = 12 + 3 = 1 + 3 = 4

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
+ 3 outside → up 3

▶ Watch it solved

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

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
+ 1 outside → up 1

▶ Watch it solved

Graph y = -x^2 from y = x^2 Harder

The question

Question: graph y = −x² from y = x².

y = −x2

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 minus in front turns every y into minus y.

So the U flips upside down.

− in front → flip

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

The turning point stays at (0, 0).

4 Draw the new graph

The shape is flipped.

From the turning point, 1 right, 1 down.

And 1 left, 1 down.

Join them with an upside-down U.

This is y = −x².

5 Check

Let's check.

Put x = 0 in the new rule.

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

x = 0: y = −02 = 0

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

x = 1: y = −12 = −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
− in front → flip

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. y = x^2 + 3 from y = x^2
    Answer
    + 3 outside → up 3
  2. y = x^2 - 4 from y = x^2
    Answer
    − 4 outside → down 4
  3. y = (x - 2)^2 from y = x^2
    Answer
    x − 2 inside → right 2
  4. y = (x + 1)^2 from y = x^2
    Answer
    x + 1 inside → left 1
  5. y = |x| + 2 from y = |x|
    Answer
    + 2 outside → up 2
  6. y = |x - 3| from y = |x|
    Answer
    x − 3 inside → right 3
  7. y = |x + 4| from y = |x|
    Answer
    x + 4 inside → left 4
  8. y = |x| - 2 from y = |x|
    Answer
    − 2 outside → down 2

Medium

  1. y = (x - 3)^2 + 1 from y = x^2
    Answer
    + 1 outside → up 1
  2. y = (x + 2)^2 - 3 from y = x^2
    Answer
    − 3 outside → down 3
  3. y = |x - 1| + 2 from y = |x|
    Answer
    + 2 outside → up 2
  4. y = |x + 3| - 1 from y = |x|
    Answer
    − 1 outside → down 1
  5. y = (x - 4)^2 - 2 from y = x^2
    Answer
    − 2 outside → down 2
  6. y = (x + 1)^2 + 4 from y = x^2
    Answer
    + 4 outside → up 4
  7. move y = |x| to y = |x - 2| - 3
    Answer
    − 3 outside → down 3

Harder

  1. y = -x^2 from y = x^2
    Answer
    − in front → flip
  2. y = -x^2 + 3 from y = x^2
    Answer
    + 3 outside → up 3
  3. y = -(x - 2)^2 from y = x^2
    Answer
    x − 2 inside → right 2
  4. y = -|x| + 4 from y = |x|
    Answer
    + 4 outside → up 4
  5. y = 2x^2 - 1 from y = x^2
    Answer
    − 1 outside → down 1
  6. move y = |x| to y = 2|x + 1|
    Answer
    x + 1 inside → left 1
  7. y = -(x + 1)^2 + 2 from y = x^2
    Answer
    + 2 outside → up 2