Linear Function Transformations

Notes based on your graph worksheets – translations of the parent function and how to read them.

Parent function: f(x) = x

1. The Parent Linear Function

This is the “starting graph” that all the others are moved or stretched from.

Parent linear function: f(x) = x

  • Goes through points like (-2, -2), (0, 0), (2, 2).
  • Has slope 1 (up 1, right 1).
  • Angle of the line never changes when we only translate (shift) it.

Domain & Range

For f(x) = x:

  • Domain (all x-values): all real numbers.
  • Range (all y-values): all real numbers.

Translations (sliding the line up/down/left/right) do not change the fact that x and y can still be any real number.

Think: Input → Output

For each x you plug into f(x) = x, you get the same y back.

xf(x)
-2-2
-1-1
00
11
22

Transformations will change how x turns into y, but the basic line shape stays the same.

2. Vertical Translations (Up & Down)

These match the “Vertical Translations” graph from your notes (g₃ and g₄).

Rule: g(x) = f(x) + k

  • If k > 0 → shift the graph up k units.
  • If k < 0 → shift the graph down |k| units.
  • Each point moves straight up or straight down.

Example (like g₃)

Let f(x) = x. Define g₃(x) = f(x) + 2 = x + 2.

Every y-value is 2 bigger than it used to be:

Original point on fNew point on g₃
(0, 0)(0, 2)
(1, 1)(1, 3)
(-1, -1)(-1, 1)

The line is exactly the same, just 2 units higher.

Example (like g₄)

Let g₄(x) = f(x) - 2 = x - 2.

Every y-value is 2 smaller than it used to be:

Original point on fNew point on g₄
(0, 0)(0, -2)
(1, 1)(1, -1)
(-1, -1)(-1, -3)

Key idea for vertical shifts

  • The x-coordinate does not change.
  • Only the y-values are affected.
  • Think: “+ k is outside the function → move output (y) up or down.”

3. Horizontal Translations (Left & Right)

These match the “Horizontal Translations” graph from your notes (g₁ and g₂).

Rule: g(x) = f(x + k)

  • If k > 0 → shift the graph left k units.
  • If k < 0 → shift the graph right |k| units.

Notice this feels backwards. That’s where the “crazy” rule comes in.

Inside is crazy

Anything that happens to the x inside the parentheses acts the opposite of what it looks like.

Inside → Horizontal Looks +k → actually left Looks −k → actually right

Example (like g₁)

Let g₁(x) = f(x + 2).

  • We replace x by x + 2, so input is “2 bigger.”
  • Graph shifts left 2 units.
Point on fMatching point on g₁
(0, 0)(-2, 0)
(2, 2)(0, 2)
(-1, -1)(-3, -1)

Everything is 2 units to the left.

Example (like g₂)

Let g₂(x) = f(x - 3).

  • We replace x by x - 3, so input is “3 smaller.”
  • Graph shifts right 3 units.
Point on fMatching point on g₂
(0, 0)(3, 0)
(1, 1)(4, 1)
(-2, -2)(1, -2)

Key idea for horizontal shifts

  • The y-coordinate stays the same.
  • Only the x-values are affected.
  • Because it’s inside, the direction is opposite:
    • f(x + k) → left k
    • f(x - k) → right k

4. Comparing Graphs (No Equation Given)

Like the worksheets where the blue graph is f and the red graph is g.

Steps to compare

  1. Match one key point. Good choices are the vertex (for quadratics) or an easy point like where the line crosses the y-axis.
  2. Count how far left/right the new point is from the old one → that tells you the horizontal shift.
  3. Count how far up/down the new point is → that tells you the vertical shift.
  4. Write it using g(x) = f(x ± k) ± c with the “inside is crazy” rule.

Example using a line

Suppose the blue line f goes through (0, 0) and the red line g goes through (2, 3), but they look parallel.

  • Horizontal: from x = 0 to x = 2 → moved right 2.
  • Vertical: from y = 0 to y = 3 → moved up 3.

So the transformation must be:

  • Right 2 → inside crazy: f(x - 2)
  • Up 3 → outside normal: + 3

Answer: g(x) = f(x - 2) + 3

5. How to Decode Any g(x)

Use this as a mini checklist whenever you see something like g(x) in terms of f(x).

Step-by-step

  1. Start from f(x). Know its basic shape and a couple of points.
  2. Look at what’s inside (x-part).
    • If you see f(x + k) or f(x - k), that’s a horizontal shift.
    • Remember: inside is crazy → direction is opposite.
  3. Look at what’s outside.
    • f(x) + k or f(x) - k → vertical shift.
    • Up if +k, down if −k.
  4. Optional: look for flips or stretches. (If you ever see a number times f(x), like -2f(x), that reflects and stretches.)

Example: g(x) = f(x + 2) - 3

  • Inside: x + 2 → horizontal shift left 2.
  • Outside: - 3 → vertical shift down 3.

So to get g from f:

  1. Move every point on f 2 units left.
  2. Then move every point 3 units down.

6. Practice Problems (with answers)

Try to answer first, then peek at the green solution.

1. The parent function is f(x) = x. The line g is the same slope but passes through the point (0, 4) instead of (0, 0).

Write g(x) in terms of f(x) and describe the transformation.

Answer: g(x) = f(x) + 4. This is a vertical shift up 4 units.

2. On the graph, the blue line f goes through (1, 1). A red line g goes through (-2, 1), and they are parallel.

Write g(x) in terms of f(x).

Shift: x changed from 1 to −2 → 3 units to the left.

Left 3 → inside crazy: g(x) = f(x + 3).

3. Suppose g(x) = f(x - 5) - 2 and the point (3, 3) is on f.

Find the coordinates of the matching point on g.

Horizontal: f(x - 5) → right 5.

Vertical: - 2 → down 2.

So (3, 3) moves to (8, 1).

4. Decide if each equation is a horizontal or vertical translation of f.

  • a) g(x) = f(x) - 7
  • b) g(x) = f(x + 4)
  • c) g(x) = f(x - 1) + 3

a) vertical, down 7

b) horizontal, left 4 (inside is crazy)

c) both: right 1 and up 3