Characteristics of linear functions worksheet - Characteristics of ... - Free Printable

Let's solve each problem on the "Characteristics of Linear Functions Practice Worksheet A" step by step, using the graphs provided.

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Problem 1

Graph Description: A straight line that crosses the y-axis at (0, 2) and the x-axis at (6, 0). It goes from top-left to bottom-right (decreasing).

#### Step-by-step Analysis:

- Domain: All real numbers since it’s a straight line extending infinitely.
- ✔ Domain: $(-\infty, \infty)$

- Range: All real numbers since it extends up and down infinitely.
- ✔ Range: $(-\infty, \infty)$

- x-intercept: Where the graph crosses the x-axis → at $x = 6$
- ✔ x-intercept: $6$

- y-intercept: Where the graph crosses the y-axis → at $y = 2$
- ✔ y-intercept: $2$

- Increasing/Decreasing/Constant:
- The line is going downward from left to right → decreasing
- ✔ Increasing: None
- ✔ Decreasing: $(-\infty, \infty)$
- ✔ Constant: None

- Slope: Use two points: $(0, 2)$ and $(6, 0)$
$$
m = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3}
$$
- ✔ Slope: $-\frac{1}{3}$

- End Behavior:
- As $x \to +\infty$, $f(x) \to -\infty$
- As $x \to -\infty$, $f(x) \to +\infty$
- ✔ As $x \to +\infty$, $f(x) \to -\infty$
- ✔ As $x \to -\infty$, $f(x) \to +\infty$

- Equation: Use slope-intercept form: $y = mx + b$
- $m = -\frac{1}{3}, b = 2$
- ✔ Equation: $f(x) = -\frac{1}{3}x + 2$

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✔ Final Answers for Problem 1:

| Characteristic | Answer |
|--------------------------|--------------------------------|
| Domain | $(-\infty, \infty)$ |
| Range | $(-\infty, \infty)$ |
| x-intercept | $6$ |
| y-intercept | $2$ |
| Increasing | None |
| Decreasing | $(-\infty, \infty)$ |
| Constant | None |
| Slope | $-\frac{1}{3}$ |
| End Behavior: $x \to +\infty$ | $f(x) \to -\infty$ |
| End Behavior: $x \to -\infty$ | $f(x) \to +\infty$ |
| Equation | $f(x) = -\frac{1}{3}x + 2$ |

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Problem 2

Graph Description: A piecewise function with multiple segments. It starts at (0, 10), increases to (10, 60), stays constant until (15, 60), then decreases to (30, 0), continues decreasing to (40, -40), then increases back to (55, 10).

⚠️ Important Note: This graph is not linear — it's piecewise and includes increasing, decreasing, and constant parts. However, the worksheet is titled "Linear Functions", so this might be an error or test of understanding.

But let's analyze based on what's shown.

#### Step-by-step Analysis:

- Domain: From $x = 0$ to $x = 55$
- ✔ Domain: $[0, 55]$

- Range: Minimum is $-40$, maximum is $60$
- ✔ Range: $[-40, 60]$

- x-intercept: Where $f(x) = 0$ → appears at $x = 30$ (from graph)
- ✔ x-intercept: $30$

- y-intercept: At $x = 0$, $y = 10$
- ✔ y-intercept: $10$

- Increasing/Decreasing/Constant:
- Increasing: $[0, 10]$ and $[40, 55]$
- Decreasing: $[15, 30]$ and $[30, 40]$
- Constant: $[10, 15]$

✔ Increasing: $[0, 10] \cup [40, 55]$
✔ Decreasing: $[15, 40]$
✔ Constant: $[10, 15]$

- Slope: Not constant → not a linear function, so no single slope.
- But if we're forced to answer, perhaps they expect “no constant slope” or “varies”
- ✔ Slope: Varies (not constant)

- End Behavior:
- As $x \to +\infty$: Function ends at $x=55$, $f(55)=10$, but doesn’t extend beyond → not defined beyond 55
- As $x \to -\infty$: Not defined below $x=0$
- So end behavior is not applicable because domain is bounded.
- ✔ As $x \to +\infty$, $f(x) \to ?$ → Not defined (domain stops at 55)
→ Undefined / Not applicable
- ✔ As $x \to -\infty$, $f(x) \to ?$ → Undefined

- Equation: Cannot write one equation; it's piecewise.
- ✔ Equation: Piecewise function (not linear)

> ⚠️ Note: This graph is not linear, so it may be a trick question or mislabeled. But we answer based on the graph.

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✔ Final Answers for Problem 2:

| Characteristic | Answer |
|--------------------------|----------------------------------------|
| Domain | $[0, 55]$ |
| Range | $[-40, 60]$ |
| x-intercept | $30$ |
| y-intercept | $10$ |
| Increasing | $[0, 10] \cup [40, 55]$ |
| Decreasing | $[15, 40]$ |
| Constant | $[10, 15]$ |
| Slope | Varies (not constant) |
| End Behavior: $x \to +\infty$ | Undefined (domain limited) |
| End Behavior: $x \to -\infty$ | Undefined (domain limited) |
| Equation | Not linear; piecewise function |

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Problem 3

Graph Description: A straight line passing through (-4, -7) and (2, 5), increasing from bottom-left to top-right.

#### Step-by-step Analysis:

- Domain: All real numbers
- ✔ Domain: $(-\infty, \infty)$

- Range: All real numbers
- ✔ Range: $(-\infty, \infty)$

- x-intercept: Where $y = 0$. Let’s find it.
- First, find equation using two points: $(-4, -7)$ and $(2, 5)$
- Slope: $m = \frac{5 - (-7)}{2 - (-4)} = \frac{12}{6} = 2$
- Use point-slope: $y - 5 = 2(x - 2)$ → $y = 2x + 1$
- Set $y = 0$: $0 = 2x + 1$ → $x = -\frac{1}{2}$
- ✔ x-intercept: $-\frac{1}{2}$

- y-intercept: When $x = 0$: $y = 2(0) + 1 = 1$
- ✔ y-intercept: $1$

- Increasing/Decreasing/Constant:
- Line has positive slope → increasing everywhere
- ✔ Increasing: $(-\infty, \infty)$
- ✔ Decreasing: None
- ✔ Constant: None

- Slope: Already calculated
- ✔ Slope: $2$

- End Behavior:
- As $x \to +\infty$, $f(x) \to +\infty$
- As $x \to -\infty$, $f(x) \to -\infty$
- ✔ As $x \to +\infty$, $f(x) \to +\infty$
- ✔ As $x \to -\infty$, $f(x) \to -\infty$

- Equation: $y = 2x + 1$
- ✔ Equation: $f(x) = 2x + 1$

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✔ Final Answers for Problem 3:

| Characteristic | Answer |
|--------------------------|--------------------------------|
| Domain | $(-\infty, \infty)$ |
| Range | $(-\infty, \infty)$ |
| x-intercept | $-\frac{1}{2}$ |
| y-intercept | $1$ |
| Increasing | $(-\infty, \infty)$ |
| Decreasing | None |
| Constant | None |
| Slope | $2$ |
| End Behavior: $x \to +\infty$ | $f(x) \to +\infty$ |
| End Behavior: $x \to -\infty$ | $f(x) \to -\infty$ |
| Equation | $f(x) = 2x + 1$ |

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📌 Summary:

Only Problem 1 and Problem 3 are linear functions.
Problem 2 is not linear — it's piecewise and should not be classified as a linear function.

However, since the worksheet asks for characteristics, we answered all accordingly.

If you need these answers formatted neatly for printing or submission, let me know!