Graph Transformations: Shifts and Vertical Scaling

Lesson 1 of 2

In this lesson:

• Shift graphs up, down, left, and right
• Stretch, compress, and reflect graphs vertically

Learning Objectives for This Lesson

1. Identify vertical translations: shifts up or down by
2. Identify horizontal translations: shifts left or right by
3. Explain why horizontal shifts oppose the sign of
4. Identify vertical stretches, compressions, and reflections:
5. Determine from graphs of and a transformation

What Changes When We Modify a Function?

You already know these parent functions and their graphs:

• → parabola opening upward
• → half-parabola (domain )
• → V-shape

When we change the formula, which part of the graph changes?

Vertical Shift: Adding Outside the Function

• adds to every output (y-value)
• → shifts the graph up by units
• → shifts the graph down by units
• x-coordinates are unchanged — only y-values move

Quick Check: Vertical Shift Direction

has its vertex at .

Where does the vertex move for each?

• : vertex at
• : vertex at

Think before advancing.

Horizontal Shift: The Surprising Direction

shifts the graph horizontally — but the direction opposes the sign:

• : shifts left by 3 (k = +3, moves left)
• : shifts right by 2 (k = −2, moves right)

The direction opposes the sign of .

Why Does Shift Left?

The vertex of occurs where , which gives .

The Inside-Outside Transformation Rule Explained

Outside: vertical, direction matches. Inside: horizontal, direction reversed.

Worked Example: Identify the Shifts

From to :

• Inside: → opposes → shift right 3
• Outside: → matches → shift up 4

Vertex moves from to .

Worked Example: Equation from Description

Graph of shifted left 2 and down 5. Write .

• Left 2 → inside, opposes sign → write (not )
• Down 5 → outside, matches sign → write

Quick Check: Find for a Right Shift

shifts the graph right by 4 units.

What is ?

The shift opposes the sign — so if the graph moves right, is...

Practice: Describe Each Graph Translation

For , describe the shift for each:

Write your descriptions, then advance.

Answers: Describing Each Graph Translation

1. : up 6 (outside, matches +6)
2. : left 5 (inside, opposes +5)
3. : right 1, down 3 (inside opposes −1; outside matches −3)
4. : down 4 (outside, matches −4)

From Graph Shifts to Vertical Scaling

So far: adding to the input or output moves the graph.

Now: multiplying the output changes its size.

Vertical Scaling: Stretch and Compress

For , every y-value is multiplied by :

• → vertical stretch (graph grows faster, appears steeper)
• → vertical compression (graph grows slower, appears wider)
• x-coordinates are unchanged throughout

Visual: Three Versions of

grows faster — reaches height 4 at . grows slower — reaches height 4 at .

Vertical Reflection: When

When in :

• Every y-value flips sign → reflection over the x-axis
• If : also stretched or compressed simultaneously

Example: → parabola opens downward

Point Tracking Through Vertical Scaling

Same x-value (), different y-values under each transformation.

Worked Example: Stretch and Reflect Together

Describe the transformation from to :

• : negative → reflection over the x-axis
• → vertical stretch by factor 2

Vertex stays at . Parabola opens downward, steeper than .

Worked Example: All Four Operations Together

Identify each transformation in :

1. Inside: → opposes sign → left 2
2. Multiplier: → reflect over x-axis + compress by 0.5
3. Outside: → matches sign → up 1

Quick Check: Classify Each Scaling Transformation

Classify each:

• : stretch or compress?
• : stretch or compress?
• : reflection over x-axis or y-axis?

Answer each before advancing.

Practice: Identifying Vertical Scaling Transformations

For , describe each (stretch/compress/reflect + factor):

1. — vertical stretch by 4
2. — vertical compression by 0.25
3. — reflection over x-axis
4. — vertical stretch by 3 + reflection over x-axis

Key Takeaways: Shifts and Vertical Scaling

✓ : vertical shift — direction matches
✓ : horizontal shift — direction opposes
✓ : vertical scale — stretch, compress, reflects

shifts left — inside opposes the sign
Outside changes y-values only; inside changes x-values only
Parse combined transforms inside-out

Coming Up: Lesson 2 Topic Preview

Next lesson: Graph Transformations — Horizontal Scaling and Symmetry

• : multiplying the input by (another surprise direction)
• : reflecting over the y-axis
• Even and odd functions: classifying graphs by symmetry

Preview question: If shifts left, what do you predict does to the graph?