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Lesson 4: Graphs of Functions & Transformations

Estimated time: 35-40 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Identify and graph parent functions
Apply vertical shifts to function graphs
Apply horizontal shifts to function graphs
Apply vertical stretches and compressions
Apply reflections over x-axis and y-axis
Combine multiple transformations in the correct order

Parent Functions

Before we can transform functions, we need to know the parent functions—the simplest form of each function family. Think of them as the "building blocks" that we'll modify using transformations.

Parent Function: The most basic form of a function family, without any transformations applied.

Here are the six essential parent functions you need to know:

1. Linear Function

f(x) = x

Graph: Diagonal line through origin with slope 1

Domain: All reals

Range: All reals

2. Quadratic Function

f(x) = x²

Graph: U-shaped parabola opening upward, vertex at origin

Domain: All reals

Range: [0, ∞)

3. Cubic Function

f(x) = x³

Graph: S-shaped curve through origin

Domain: All reals

Range: All reals

4. Absolute Value Function

f(x) = |x|

Graph: V-shape with vertex at origin

Domain: All reals

Range: [0, ∞)

5. Square Root Function

f(x) = √x

Graph: Half-parabola starting at origin, opening right

Domain: [0, ∞)

Range: [0, ∞)

6. Reciprocal Function

f(x) = 1/x

Graph: Hyperbola in quadrants I and III

Domain: All reals except 0

Range: All reals except 0

Pro Tip

Memorize these six parent functions! You'll use them constantly in algebra, and being able to sketch them quickly will save you tons of time on homework and exams.

Vertical Shifts

The simplest transformation is a vertical shift—moving the entire graph up or down.

Vertical Shift: f(x) + k

If k > 0: Shift UP k units
If k < 0: Shift DOWN k units

Examples: Vertical Shifts

Parent function: f(x) = x²

1. g(x) = x² + 3

Shift the parabola UP 3 units
Vertex moves from (0, 0) to (0, 3)

2. h(x) = x² − 5

Shift the parabola DOWN 5 units
Vertex moves from (0, 0) to (0, −5)

3. k(x) = |x| + 2

Shift the V-shape UP 2 units
Vertex moves from (0, 0) to (0, 2)

Horizontal Shifts

Horizontal shifts move the graph left or right. Watch out—the direction is OPPOSITE of what you might expect!

Horizontal Shift: f(x − h)

If h > 0: Shift RIGHT h units (opposite of the sign!)
If h < 0: Shift LEFT |h| units

This is Tricky!

f(x − 2) shifts RIGHT 2 (not left!)
f(x + 3) shifts LEFT 3 (not right!)

Think of it this way: To make x − 2 equal zero, you need x = 2 (right of origin).
To make x + 3 equal zero, you need x = −3 (left of origin).

Examples: Horizontal Shifts

Parent function: f(x) = x²

1. g(x) = (x − 2)²

Shift the parabola RIGHT 2 units
Vertex moves from (0, 0) to (2, 0)

2. h(x) = (x + 4)²

Shift the parabola LEFT 4 units
Vertex moves from (0, 0) to (−4, 0)

3. k(x) = |x − 1|

Shift the V-shape RIGHT 1 unit
Vertex moves from (0, 0) to (1, 0)

Vertical Stretch and Compression

Stretches and compressions change the "width" or "steepness" of a graph by multiplying the function by a constant.

Vertical Stretch/Compression: a · f(x)

If |a| > 1: Vertical STRETCH (graph becomes taller, narrower)
If 0 < |a| < 1: Vertical COMPRESSION (graph becomes shorter, wider)

Examples: Vertical Stretch/Compression

Parent function: f(x) = x²

1. g(x) = 3x²

Vertical STRETCH by factor of 3
Graph becomes narrower (steeper)
When x = 2: f(2) = 4, but g(2) = 12 (3 times taller)

2. h(x) = (1/2)x²

Vertical COMPRESSION by factor of 1/2
Graph becomes wider (less steep)
When x = 2: f(2) = 4, but h(2) = 2 (half as tall)

3. k(x) = 4|x|

Vertical STRETCH by factor of 4
V-shape becomes narrower

Reflections

Reflections "flip" the graph over an axis, creating a mirror image.

Reflections:

−f(x): Reflect over the x-axis (flip upside down)
f(−x): Reflect over the y-axis (flip left to right)

Examples: Reflections

Parent function: f(x) = x²

1. g(x) = −x²

Reflect over x-axis
Parabola now opens DOWNWARD
Vertex still at (0, 0), but graph is flipped

2. h(x) = (−x)² = x²

Reflect over y-axis
For x², this looks the same (parabola is symmetric)
But try with f(x) = x³: f(−x) = (−x)³ = −x³ (different!)

3. k(x) = −√x

Reflect √x over x-axis
Graph now points downward instead of upward

Memory Trick

−f(x): Negative outside → affects the output → flips over x-axis
f(−x): Negative inside → affects the input → flips over y-axis

Combining Multiple Transformations

Real-world problems often involve multiple transformations at once. The key is applying them in the correct order.

Order of Transformations (CRITICAL!):

Horizontal shifts (inside parentheses)
Reflections and stretches/compressions
Vertical shifts (added/subtracted outside)

Transformation	Form	Effect
Horizontal Shift	f(x − h)	Right h (if h > 0), Left \|h\| (if h < 0)
Vertical Shift	f(x) + k	Up k (if k > 0), Down \|k\| (if k < 0)
Vertical Stretch	a · f(x), \|a\| > 1	Taller, narrower
Vertical Compression	a · f(x), 0 < \|a\| < 1	Shorter, wider
Reflect over x-axis	−f(x)	Flip upside down
Reflect over y-axis	f(−x)	Flip left to right

Example: Multi-Step Transformation

Describe all transformations: g(x) = −2(x − 3)² + 1

Starting from parent function f(x) = x²

Step-by-step breakdown:

Step 1: Horizontal shift

(x − 3)² → Shift RIGHT 3 units

Step 2: Vertical stretch

2(x − 3)² → Stretch by factor of 2

Step 3: Reflection over x-axis

−2(x − 3)² → Flip over x-axis (opens downward)

Step 4: Vertical shift

−2(x − 3)² + 1 → Shift UP 1 unit

Final result:

Vertex at (3, 1)
Opens downward
Narrower than parent function

Example: Writing the Equation

Write the equation for f(x) = |x| with these transformations:

Shift right 4 units
Shift down 2 units
Reflect over x-axis

Solution:

Start: f(x) = |x|

Shift right 4: |x − 4|

Reflect over x-axis: −|x − 4|

Shift down 2: −|x − 4| − 2

Answer: g(x) = −|x − 4| − 2

Check Your Understanding

Try these questions to see if you've grasped the key concepts:

1. What transformation does g(x) = x² + 5 represent?

Answer: Vertical shift UP 5 units. The +5 is added outside the function, so it moves the entire graph upward by 5 units. The vertex moves from (0, 0) to (0, 5).

2. Does g(x) = (x − 2)² shift left or right? By how many units?

Answer: Shift RIGHT 2 units. Remember: horizontal shifts are opposite of the sign! (x − 2) means move right 2. The vertex moves from (0, 0) to (2, 0).

3. How does g(x) = 3x² differ from f(x) = x²?

Answer: Vertical STRETCH by factor of 3. The coefficient 3 (where |3| > 1) makes the parabola narrower and steeper. Every y-value is multiplied by 3, making the graph taller.

4. What does the negative sign do in g(x) = −x²?

Answer: Reflection over the x-axis. The negative sign outside the function flips the graph upside down. The parabola now opens downward instead of upward.

5. Describe all transformations in g(x) = −(x + 1)² − 3.

Answer: Starting from f(x) = x²:
1. Shift LEFT 1 unit (x + 1)
2. Reflect over x-axis (negative sign)
3. Shift DOWN 3 units (−3 outside)

Final result: Vertex at (−1, −3), opens downward

Key Takeaways

Know the six parent functions: linear, quadratic, cubic, absolute value, square root, reciprocal
Vertical shifts: f(x) + k → up if k > 0, down if k < 0
Horizontal shifts: f(x − h) → RIGHT if h > 0 (opposite of sign!), LEFT if h < 0
Vertical stretch: a·f(x) where |a| > 1 → taller, narrower
Vertical compression: a·f(x) where 0 < |a| < 1 → shorter, wider
Reflection over x-axis: −f(x) → flip upside down
Reflection over y-axis: f(−x) → flip left to right
Order matters: Horizontal shifts → Stretches/Reflections → Vertical shifts

Congratulations! You've Completed Module 1!

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