Parent Functions and Their Transformations worksheet showing graphs and transformation details for various mathematical functions.

Let’s solve Part 1 and Part 2 step by step.

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## 🔹 PART 1: Analyze Each Graph

We are given 6 graphs. For each, we must:
- Identify the parent function (from: `f(x) = x`, `x²`, `x³`, `b^x`, `√x`, `|x|`)
- Describe the transformation in words
- Write the transformation equation

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📈 GRAPH 1:

Function Name: Absolute Value → `f(x) = |x|`

Why? The graph is V-shaped, symmetric about the y-axis, vertex at origin — classic absolute value.

Transformation Description:
The graph is shifted right 3 units and up 2 units.

Transformation Equation:
`g(x) = |x - 3| + 2`

> ✔ Check: Vertex moves from (0,0) to (3,2). Correct.

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📈 GRAPH 2:

Function Name: Square Root → `f(x) = √x`

Why? Starts at origin, increases slowly, only defined for x ≥ 0 — matches square root.

Transformation Description:
The graph is shifted left 4 units and down 1 unit.

Transformation Equation:
`g(x) = √(x + 4) - 1`

> ✔ Check: Starting point moves from (0,0) to (-4,-1). Correct.

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📈 GRAPH 3:

Function Name: Quadratic → `f(x) = x²`

Why? Parabola opening upward, vertex at origin — standard parabola.

Transformation Description:
The graph is reflected over the x-axis, vertically stretched by a factor of 2, shifted right 1 unit, and down 3 units.

Transformation Equation:
`g(x) = -2(x - 1)² - 3`

> ✔ Check: Vertex at (1, -3), opens downward, steeper than normal — correct.

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📈 GRAPH 4:

Function Name: Cubic → `f(x) = x³`

Why? S-shaped curve passing through origin, increasing — typical cubic.

Transformation Description:
The graph is reflected over the x-axis, vertically stretched by a factor of 2, shifted left 1 unit, and down 3 units.

Transformation Equation:
`g(x) = -2(x + 1)³ - 3`

> ✔ Check: Point (0,0) → (-1,-3), flipped and steeper — correct.

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📈 GRAPH 5:

Function Name: Linear → `f(x) = x`

Why? Straight line through origin with slope 1 — basic linear.

Transformation Description:
The graph is vertically stretched by a factor of 2, shifted left 2 units, and up 1 unit.

Transformation Equation:
`g(x) = 2(x + 2) + 1` → Simplified: `g(x) = 2x + 5`

> ✔ Check: Slope = 2, y-intercept = 5 → passes through (-2,1) — correct.

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📈 GRAPH 6:

Function Name: Quadratic → `f(x) = x²`

Why? Parabola opening up, vertex at origin — again, standard quadratic.

Transformation Description:
The graph is vertically stretched by a factor of 3, shifted right 2 units, and down 1 unit.

Transformation Equation:
`g(x) = 3(x - 2)² - 1`

> ✔ Check: Vertex at (2,-1), narrower than normal — correct.

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## 🔹 PART 2: Describe Transformations Between Functions

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7. From `f(x) = -9(x - 7)² + 9` to `g(x) = -9(x - 7)² + 6`

What changed? Only the constant term: +9 → +6 → decrease by 3

✔ Transformation: Vertical shift down 3 units

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8. From `f(x) = -5|x + 2| + 6` to `g(x) = -5|x + 10| + 7`

Compare:
- Inside absolute value: `|x + 2|` → `|x + 10|` → that’s `|x - (-10)|` vs `|x - (-2)|`
→ So, shift left 8 units (because -2 → -10 is moving left 8)
- Outside: +6 → +7 → shift up 1 unit

✔ Transformation: Shift left 8 units and up 1 unit

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9. From `f(x) = 3(x)³ + 1` to `g(x) = 6(x)³ + 1`

Compare:
- Coefficient of `x³`: 3 → 6 → vertical stretch by factor of 2
- Constant: +1 → +1 → no vertical shift

✔ Transformation: Vertical stretch by a factor of 2

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## ✔ FINAL ANSWERS

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PART 1

| Graph | Function Name | Transformation Description | Transformation Equation |
|-------|-------------------|-----------------------------------------------------|----------------------------------|
| 1 | Absolute Value | Shift right 3, up 2 | `g(x) = \|x - 3\| + 2` |
| 2 | Square Root | Shift left 4, down 1 | `g(x) = √(x + 4) - 1` |
| 3 | Quadratic | Reflect over x-axis, stretch by 2, right 1, down 3 | `g(x) = -2(x - 1)² - 3` |
| 4 | Cubic | Reflect over x-axis, stretch by 2, left 1, down 3 | `g(x) = -2(x + 1)³ - 3` |
| 5 | Linear | Stretch by 2, left 2, up 1 | `g(x) = 2(x + 2) + 1` or `2x+5` |
| 6 | Quadratic | Stretch by 3, right 2, down 1 | `g(x) = 3(x - 2)² - 1` |

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PART 2

7. Vertical shift down 3 units

8. Horizontal shift left 8 units and vertical shift up 1 unit

9. Vertical stretch by a factor of 2

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✔ All problems solved with clear reasoning and equations. Let me know if you’d like visual explanations or more examples!