Algebra 2: Section 1.1 - Parent Functions and Transformations - Free Printable

Looking at the image, you’ve uploaded a set of 8 graphs (labeled a–h), each with handwritten domain (D) and range (R) annotations. Your task is to verify whether the given domain and range for each graph are correct, and if not, correct them.

Let’s go through each one carefully.

---

Graph a (top left — V-shaped, absolute value-like)

- Given: D: (-∞, ∞), R: [0, ∞)
- Analysis: The graph appears to be y = |x|, which is defined for all real x (so domain is all reals). The output (y-values) starts at 0 and goes up to infinity → range [0, ∞).
- ✔ Correct

---

Graph b (top middle — square root shape starting at origin)

- Given: D: [0, ∞), R: [0, ∞)
- Analysis: This looks like y = √x. Domain is x ≥ 0 (since square root of negative numbers isn’t real). Range is also y ≥ 0.
- ✔ Correct

---

Graph c (top right — horizontal line at y=1)

- Given: D: (-∞, ∞), R: {1} (written as [1] — probably meant to be {1})
- Analysis: Horizontal line means constant function y = 1. Defined for all x → domain (-∞, ∞). Only output is 1 → range is {1}.
- ❗ Almost correct — notation “R: [1]” is misleading. It should be R: {1} (a set with one element), not an interval. If they meant singleton set, it’s acceptable colloquially, but technically incorrect.
- 🟡 Needs correction: R: {1}

---

Graph d (middle left — exponential growth from left to right, asymptotic to x-axis)

- Given: D: (-∞, ∞), R: (0, ∞)
- Analysis: Looks like y = e^x or similar exponential function. Defined for all real x → domain (-∞, ∞). Asymptote at y=0, never reaches 0 → range (0, ∞).
- ✔ Correct

---

Graph e (middle center — cubic-like, passes through origin, increasing)

- Given: *No handwritten D/R provided*
- Analysis: Appears to be y = x³ or similar odd-degree polynomial. Defined for all real x → D: (-∞, ∞). Output covers all real y → R: (-∞, ∞).
- 🟢 Missing annotation — should be:
- D: (-∞, ∞)
- R: (-∞, ∞)

---

Graph f (middle right — straight line through origin, slope 1)

- Given: *No handwritten D/R provided*
- Analysis: Looks like y = x. Defined for all x → D: (-∞, ∞). Output covers all y → R: (-∞, ∞).
- 🟢 Missing annotation — should be:
- D: (-∞, ∞)
- R: (-∞, ∞)

---

Graph g (bottom left — hyperbola, two branches in QI and QIII)

- Given: *No handwritten D/R provided*
- Analysis: Looks like y = 1/x. Undefined at x=0 → D: (-∞, 0) ∪ (0, ∞). Never equals 0 → R: (-∞, 0) ∪ (0, ∞).
- 🟢 Missing annotation — should be:
- D: (-∞, 0) ∪ (0, ∞)
- R: (-∞, 0) ∪ (0, ∞)

---

Graph h (bottom right — parabola opening upward, vertex at origin)

- Given: *No handwritten D/R provided*
- Analysis: Looks like y = x². Defined for all real x → D: (-∞, ∞). Minimum y-value is 0 → R: [0, ∞).
- 🟢 Missing annotation — should be:
- D: (-∞, ∞)
- R: [0, ∞)

---

## ✔ Final Summary:

| Graph | Given D/R | Correct? | Correction Needed |
|-------|-----------|----------|-------------------|
| a | D: (-∞,∞), R: [0,∞) | ✔ Yes | None |
| b | D: [0,∞), R: [0,∞) | ✔ Yes | None |
| c | D: (-∞,∞), R: [1] | ⚠️ Almost | R: {1} |
| d | D: (-∞,∞), R: (0,∞) | ✔ Yes | None |
| e | — | ✘ Missing | D: (-∞,∞), R: (-∞,∞) |
| f | — | ✘ Missing | D: (-∞,∞), R: (-∞,∞) |
| g | — | ✘ Missing | D: (-∞,0)∪(0,∞), R: (-∞,0)∪(0,∞) |
| h | — | ✘ Missing | D: (-∞,∞), R: [0,∞) |

---

## 💡 Key Concepts Used:

- Domain: All possible x-values where the function is defined.
- Range: All possible y-values the function can output.
- For continuous functions (polynomials, exponentials, etc.), check endpoints and asymptotes.
- For piecewise or restricted functions (like √x, 1/x), watch for undefined points or restrictions.
- Use interval notation correctly: parentheses ( ) for open, brackets [ ] for closed, curly braces { } for single elements.

---

✔ You now have a complete, corrected analysis of all 8 graphs. Let me know if you’d like visual explanations or further breakdowns!