1. **Problem statement:** Determine the type of angle (acute or obtuse) formed by the part of the graph above the x-axis with the positive x-axis direction, and whether the function is increasing, decreasing, or constant for each linear function given.
2. **Recall:** For a linear function $y = mx + b$, the slope $m$ determines the angle and monotonicity:
- If $m > 0$, the angle with the positive x-axis is acute, and the function is increasing.
- If $m < 0$, the angle is obtuse, and the function is decreasing.
- If $m = 0$, the function is constant and the angle is $0^\circ$ (horizontal line).
3. **Analyze each function:**
(2) $y = -2x + 1$:
- Slope $m = -2 < 0$.
- Angle: obtuse.
- Function: decreasing.
(4) $y + x = 9 \Rightarrow y = -x + 9$:
- Slope $m = -1 < 0$.
- Angle: obtuse.
- Function: decreasing.
(6) $y = 7$:
- Slope $m = 0$.
- Angle: $0^\circ$ (horizontal).
- Function: constant.
(8) $y = -x$:
- Slope $m = -1 < 0$.
- Angle: obtuse.
- Function: decreasing.
(10) $2x + y = 3 \Rightarrow y = -2x + 3$:
- Slope $m = -2 < 0$.
- Angle: obtuse.
- Function: decreasing.
(12) $y = 7x + 7$:
- Slope $m = 7 > 0$.
- Angle: acute.
- Function: increasing.
(14) $-3.2x = y + 1 \Rightarrow y = -3.2x - 1$:
- Slope $m = -3.2 < 0$.
- Angle: obtuse.
- Function: decreasing.
(16) $y = 4 - \frac{2}{5}x$:
- Slope $m = -\frac{2}{5} < 0$.
- Angle: obtuse.
- Function: decreasing.
(18) $y = \frac{9 - 3x}{3} = 3 - x$:
- Slope $m = -1 < 0$.
- Angle: obtuse.
- Function: decreasing.
(1) $y = 5x - 8$:
- Slope $m = 5 > 0$.
- Angle: acute.
- Function: increasing.
(3) $y = -0.3x - 10$:
- Slope $m = -0.3 < 0$.
- Angle: obtuse.
- Function: decreasing.
(5) $y = -\frac{3}{4}x + 3$:
- Slope $m = -\frac{3}{4} < 0$.
- Angle: obtuse.
- Function: decreasing.
(7) $y = 2 + 3x$:
- Slope $m = 3 > 0$.
- Angle: acute.
- Function: increasing.
(9) $y = \frac{5x}{7}$:
- Slope $m = \frac{5}{7} > 0$.
- Angle: acute.
- Function: increasing.
(11) $y = -4 - \frac{1}{2}x$:
- Slope $m = -\frac{1}{2} < 0$.
- Angle: obtuse.
- Function: decreasing.
(13) $y - x = 6 - x \Rightarrow y = 6$:
- Slope $m = 0$.
- Angle: $0^\circ$ (horizontal).
- Function: constant.
(15) $y + 5x = 3 \Rightarrow y = -5x + 3$:
- Slope $m = -5 < 0$.
- Angle: obtuse.
- Function: decreasing.
(17) $y = \frac{-2x + 5}{7} = -\frac{2}{7}x + \frac{5}{7}$:
- Slope $m = -\frac{2}{7} < 0$.
- Angle: obtuse.
- Function: decreasing.
**Summary:**
- Acute angle and increasing: (1), (7), (9), (12)
- Obtuse angle and decreasing: (2), (3), (4), (5), (8), (10), (11), (14), (15), (16), (17), (18)
- Constant function (horizontal line): (6), (13)
Angle And Monotonicity
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