Algebra 1 - Interpreting Graphs of Functions - Free Printable

The image you've shared is a worksheet titled "Interpreting Graphs of Functions" from @iTEACHAL9EBRA, designed to help students analyze and interpret different types of graphs. It includes four real-world scenarios represented by graphs, each followed by questions about the function's characteristics.

Let’s go through each graph and solve/analyze the problems step by step.

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🔹 Graph 1: David’s Savings for Car

- Graph: A straight line increasing from $1400 at week 0 to $2200 at week 10.
- Type: Linear (straight line)

#### Questions:
- Linear or nonlinear?
✔ Linear

- x-int:
The x-intercept is where savings = 0. But the graph starts at $1400 at week 0 and increases. So it never reaches $0.
→ None (or not in domain shown)

- y-int:
At week 0, savings = $1400 → 1400

- Is there line symmetry?
No, because it's a straight line with constant slope — only symmetric if horizontal or vertical, which it isn't.
→ No

- Positive/negative intervals:
Since savings are always positive and increasing:
- Positive: all x-values (since y > 0)
- Negative: none

- Increasing/decreasing:
- Increasing: for all x in [0,10] → 0 < x < 10
- Decreasing: none

- Relative extrema (x-value):
No peaks or valleys → none

✔ Summary:
- Linear
- y-int: 1400
- x-int: none
- Symmetry: no
- Increasing: 0 < x < 10
- No decreasing
- No relative extrema

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🔹 Graph 2: Height of Golf Ball

- Graph: Parabola opening downward, peak at (60, 160), hits ground at x = 0 and x = 120
- Type: Nonlinear (quadratic)

#### Questions:
- Linear or nonlinear?
✔ Nonlinear

- x-int:
Where height = 0 → at x = 0 and x = 120 → 0 and 120

- y-int:
At x = 0, height = 0 → 0

- Is there line symmetry?
Yes! This is a parabola → symmetric about vertex
→ Yes, x = 60

- Positive/negative intervals:
- Positive: when ball is in air → 0 < x < 120
- Negative: none (height can't be negative here)

- Increasing/decreasing:
- Increasing: from tee to peak → 0 < x < 60
- Decreasing: from peak to landing → 60 < x < 120

- Relative extrema (x-value):
Maximum at vertex → x = 60

✔ Summary:
- Nonlinear
- x-int: 0 and 120
- y-int: 0
- Symmetry: yes, x = 60
- Positive: 0 < x < 120
- Increasing: 0 < x < 60
- Decreasing: 60 < x < 120
- Relative max at x = 60

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🔹 Graph 3: Right Whale Population

- Graph: Straight line decreasing from (0, 240) to (12, 80)
- Type: Linear

#### Questions:
- Linear or nonlinear?
✔ Linear

- x-int:
Where population = 0 → extrapolate: line goes from (0,240) to (12,80).
Slope = (80 - 240)/(12 - 0) = -160/12 = -40/3 ≈ -13.33
Set y = 0:
0 = -40/3 x + 240 → x = (240 × 3)/40 = 18
→ x = 18 (not on graph, but mathematically correct)

But since the graph stops at x=12, we may say none within shown domain, but if extended, x-int = 18

However, based on the handwritten answer, they wrote 10 — that seems incorrect.

Let's check: If x-int = 10, then at x=10, y=0. But at x=12, y=80. That would mean it's rising — contradiction.

✘ So handwritten "10" is wrong. Correct x-int = 18

- y-int:
At x=0, y=240 → 240

- Symmetry?
No — straight line not symmetric unless horizontal → No

- Positive/negative:
Population is always positive in this range →
- Positive: all x in [0,12]
- Negative: none

- Increasing/decreasing:
- Increasing: none
- Decreasing: x > 0 (entire domain)

- Relative extrema:
No turning point → none

✔ Summary:
- Linear
- x-int: 18 (corrected)
- y-int: 240
- Symmetry: no
- Positive: all x in domain
- Decreasing: x > 0
- No extrema

> ⚠️ Note: The student wrote x-int = 10 — this is incorrect.

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🔹 Graph 4: Gateway Arch

- Graph: Parabola opening downward, vertex at (0, 640), x-intercepts at (-320, 0) and (320, 0)
- Type: Nonlinear

#### Questions:
- Linear or nonlinear?
✔ Nonlinear

- x-int:
Where height = 0 → -320 and 320

- y-int:
At x = 0, height = 640 → 640

- Symmetry?
Yes — parabola symmetric about vertex → Yes, x = 0

- Positive/negative:
Height is positive between the roots →
- Positive: -320 < x < 320
- Negative: none (height can’t be negative)

- Increasing/decreasing:
- Increasing: left side → -320 < x < 0
- Decreasing: right side → 0 < x < 320

- Relative extrema:
Maximum at vertex → x = 0

✔ Summary:
- Nonlinear
- x-int: -320 and 320
- y-int: 640
- Symmetry: yes, x = 0
- Positive: -320 < x < 320
- Increasing: -320 < x < 0
- Decreasing: 0 < x < 320
- Relative max at x = 0

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✔ Final Answers Recap:

| Graph | Type | x-int | y-int | Symmetry? | x of symmetry | Increasing | Decreasing | Extrema |
|-------|------------|----------------|--------|-----------|---------------|------------|------------|---------|
| 1 | Linear | None | 1400 | No | – | 0 < x < 10 | None | None |
| 2 | Nonlinear | 0, 120 | 0 | Yes | 60 | 0 < x < 60 | 60 < x < 120 | x=60 (max) |
| 3 | Linear | 18 (not 10)| 240 | No | – | None | x > 0 | None |
| 4 | Nonlinear | -320, 320 | 640 | Yes | 0 | -320 < x < 0 | 0 < x < 320 | x=0 (max) |

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📝 Key Takeaways:

- Linear functions have constant rate of change, straight lines.
- Nonlinear (e.g., parabolas) have changing rates, curved graphs.
- Symmetry exists in parabolas (axis of symmetry).
- Intervals of increase/decrease depend on slope direction.
- Extrema occur at peaks/troughs (relative max/min).

Let me know if you'd like a printable version or practice problems!