Features of Function Mini Lesson 1 worksheet - Free Printable

Let's analyze Figure 1 and Figure 2 one at a time, determine whether each is a function, and identify all the key features of each graph.

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🔍 Figure 1

#### 📌 Graph Description:
- A piecewise graph with three segments:
1. A line from (-3, 4) to (1, -1)
2. A line from (1, -1) to (2, 3)
3. A horizontal line from (2, 3) to (4, 3)

All points are connected with solid lines and filled endpoints.

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#### ✔ Is it a function?

Yes, it is a function.

✔ Justification: Use the Vertical Line Test — no vertical line intersects the graph more than once. Every x-value has only one corresponding y-value.

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#### 🔎 Key Features:

| Feature | Answer |
|--------|--------|
| Function or Not a Function | Function |
| Domain | All x-values from -3 to 4 → $[-3, 4]$ |
| Range | All y-values from -1 to 4 → $[-1, 4]$ |
| Y-Intercept | Where $x = 0$: On the first segment from (-3,4) to (1,-1).<br>Find equation: slope = $\frac{-1 - 4}{1 - (-3)} = \frac{-5}{4} = -1.25$<br>Equation: $y - 4 = -1.25(x + 3)$<br>At $x = 0$: $y = 4 - 1.25(3) = 4 - 3.75 = 0.25$<br>So, (0, 0.25) or $y = \frac{1}{4}$ |
| Relative Min | At $x = 1$, $y = -1$ → (1, -1) |
| Relative Max | At $x = 2$, $y = 3$ → (2, 3) |
| Absolute Min | The lowest point on the graph is $y = -1$ at $x = 1$ → (1, -1) |
| Absolute Max | The highest point is $y = 4$ at $x = -3$ → (-3, 4) |
| Continuous or Discrete | Continuous — the graph is connected without breaks |

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✔ Summary for Figure 1:

- Function: Yes
- Domain: $[-3, 4]$
- Range: $[-1, 4]$
- Y-intercept: $(0, 0.25)$
- Relative Min: $(1, -1)$
- Relative Max: $(2, 3)$
- Absolute Min: $(1, -1)$
- Absolute Max: $(-3, 4)$
- Type: Continuous

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🔍 Figure 2

#### 📌 Graph Description:
- A smooth, continuous curve resembling a quartic (degree 4) polynomial.
- It has two "valleys" (minima) and one "peak" (maximum).
- Arrows at both ends indicate it continues infinitely in both directions.
- Points marked at intersections with axes:
- X-intercepts: $x = -2, -1, 1, 2$
- Y-intercept: $y = 4$

The graph appears symmetric about the y-axis? Let’s check:
- Yes! It looks like an even function.

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#### ✔ Is it a function?

Yes, it is a function.

✔ Justification: Passes the Vertical Line Test — every vertical line hits the graph at most once.

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#### 🔎 Key Features:

| Feature | Answer |
|--------|--------|
| Function or Not a Function | Function |
| Domain | All real numbers → $(-\infty, \infty)$ |
| Range | The lowest points are at $y = -3$, and it goes up to infinity → $[-3, \infty)$ |
| Y-Intercept | At $x = 0$, $y = 4$ → (0, 4) |
| Relative Min | Two local minima: at $x = -1.5$ and $x = 1.5$? Wait — let's see:<br>From graph: peaks and valleys at approximately:<br>- Local min at $x = -2$? No — wait.<br>Actually, from the graph: <br>It crosses x-axis at $x = -2, -1, 1, 2$.<br>And there are two dips below x-axis: one around $x = -1.5$ and $x = 1.5$?<br>Wait — look carefully:<br>The graph touches the x-axis at $x = -2, -1, 1, 2$, but also dips down to $y = -3$ at two places.<br>So:<br>- Relative min at $x = -1.5$ and $x = 1.5$? But wait — actually, looking at the shape:<br>There is a local minimum at $x = -1.5$ (but not exactly), but better to estimate from graph:<br>From visual inspection:<br>• Local min at $x \approx -1.5$, $y = -3$<br>• Local min at $x \approx 1.5$, $y = -3$<br>But the graph shows two distinct valleys at $x = -1.5$ and $x = 1.5$, both at $y = -3$<br>So: Relative Min: $(-1.5, -3)$ and $(1.5, -3)$ |
| Relative Max | One peak at $x = 0$, $y = 4$ → (0, 4) |
| Absolute Min | The lowest y-value is $y = -3$ at both $x = -1.5$ and $x = 1.5$ → (-1.5, -3) and (1.5, -3) |
| Absolute Max | There is no maximum because the graph goes up to infinity as $x \to \pm\infty$ → None / DNE |
| Continuous or Discrete | Continuous — no breaks, smooth curve |

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⚠️ Clarification:
- Absolute Max: Does not exist because the function increases without bound.
- Relative Max: Only one at $x = 0$, $y = 4$
- Relative Min: Two at $x = -1.5$ and $x = 1.5$, both at $y = -3$

Note: Since the graph extends infinitely upward, there is no absolute maximum.

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✔ Summary for Figure 2:

- Function: Yes
- Domain: $(-\infty, \infty)$
- Range: $[-3, \infty)$
- Y-intercept: $(0, 4)$
- Relative Min: $(-1.5, -3)$ and $(1.5, -3)$
- Relative Max: $(0, 4)$
- Absolute Min: $(-1.5, -3)$ and $(1.5, -3)$
- Absolute Max: Does not exist (DNE)
- Type: Continuous

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✔ Final Answers:

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#### Figure 1

| Feature | Answer |
|--------|--------|
| Function or Not a Function | Function |
| Domain | $[-3, 4]$ |
| Range | $[-1, 4]$ |
| Y-Intercept | $(0, 0.25)$ |
| Relative Min | $(1, -1)$ |
| Relative Max | $(2, 3)$ |
| Absolute Min | $(1, -1)$ |
| Absolute Max | $(-3, 4)$ |
| Continuous or Discrete | Continuous |

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#### Figure 2

| Feature | Answer |
|--------|--------|
| Function or Not a Function | Function |
| Domain | $(-\infty, \infty)$ |
| Range | $[-3, \infty)$ |
| Y-Intercept | $(0, 4)$ |
| Relative Min | $(-1.5, -3)$, $(1.5, -3)$ |
| Relative Max | $(0, 4)$ |
| Absolute Min | $(-1.5, -3)$, $(1.5, -3)$ |
| Absolute Max | Does not exist (DNE) |
| Continuous or Discrete | Continuous |

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✔ Note: For Figure 2, if you're required to give exact values, note that the x-values of minima may be approximated based on grid. If the graph is symmetric and the zeros are at $-2, -1, 1, 2$, then the minima are likely at $x = -1.5$ and $x = 1.5$, which is reasonable.

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