QUESTION IMAGE
Question
family of lines
separate the graph paper into 4 sections and then graph each family on the same graph.
| family | equation | slope | y - intercept | similarities | differences | graph and equation observations |
|---|---|---|---|---|---|---|
| a | $y = 3x + 4$ | similar | different | 1. parallel (all) | ||
| a | $y = 3x - 3$ | slopes are 3 (all) | y - intercepts | 2. positive (all) | ||
| a | $y = 3x + 1$ | 3. slopes are 3 (all) | ||||
| family b | equation | slope | y - intercept | similarities | differences | observation |
| b | $y = 2x + 1$ | 2 | 1 | 1. they all intersect at (1, 3)? | ||
| b | $y = \frac{3}{5}x + 1$ | $\frac{3}{5}$ | 1 | y - intercepts | slope | 2. |
| b | $y = -\frac{1}{4}x + 1$ | $-\frac{1}{4}$ | 1 | 3. | ||
| b | $y = -6x + 1$ | -6 | 1 | |||
| family c | equation | slope | y - intercept | similarities | differences | observation |
| c | $y = -x - 3$ | -1 | -3 | slope | y - intercept | 1. |
| c | $y = -x + 4$ | -1 | 4 | 2. | ||
| c | $y = -x$ | -1 | 0 | 3. | ||
| c | $y = -x + 1$ | -1 | 1 | |||
| family d | equation | slope | y - intercept | similarities | differences | observation |
| d | $y = x - 5$ | 1 | -5 | y - intercept | slope | 1. |
| d | $y = -2x - 5$ | -2 | -5 | 2. | ||
| d | $y = \frac{2}{5}x - 5$ | $\frac{2}{5}$ | -5 | 3. | ||
| d | $y = 3x - 5$ | 3 | -5 |
Step1: Analyze Family A Equations
Family A has equations \( y = 3x \), \( y = 3x + 4 \), \( y = 3x - 3 \), \( y = 3x + 1 \). The slope-intercept form is \( y = mx + b \), where \( m \) is slope and \( b \) is y-intercept. For these, \( m = 3 \) (same slope), \( b \) varies (0, 4, -3, 1).
Step2: Analyze Family B Equations
Family B: \( y = 2x + 1 \), \( y=\frac{3}{5}x + 1 \), \( y = -\frac{1}{4}x + 1 \), \( y = -6x + 1 \). Here, \( b = 1 \) (same y-intercept), \( m \) varies (2, \( \frac{3}{5} \), \( -\frac{1}{4} \), -6).
Step3: Analyze Family C Equations
Family C: \( y = -x - 3 \), \( y = -x + 4 \), \( y = -x \), \( y = -x + 1 \). \( m = -1 \) (same slope), \( b \) varies (-3, 4, 0, 1).
Step4: Analyze Family D Equations
Family D: \( y = x - 5 \), \( y = -2x - 5 \), \( y=\frac{2}{5}x - 5 \), \( y = 3x - 5 \). \( b = -5 \) (same y-intercept), \( m \) varies (1, -2, \( \frac{2}{5} \), 3).
Step5: Summarize Similarities/Differences
- Similarities:
- Family A: Same slope (\( m = 3 \)), different y-intercepts.
- Family B: Same y-intercept (\( b = 1 \)), different slopes.
- Family C: Same slope (\( m = -1 \)), different y-intercepts.
- Family D: Same y-intercept (\( b = -5 \)), different slopes.
- Differences:
- Family A (slope same, y-intercept different) vs Family B (y-intercept same, slope different); same for C (slope same) vs D (y-intercept same).
- Observations: Lines with same slope are parallel (Family A, C); lines with same y-intercept intersect at y-axis (Family B, D).
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- Family A: Similarity - Same slope (\( m = 3 \)); Difference - Different y - intercepts. Observation - Parallel lines (same slope).
- Family B: Similarity - Same y - intercept (\( b = 1 \)); Difference - Different slopes. Observation - Intersect at \( (0,1) \) (same y - intercept).
- Family C: Similarity - Same slope (\( m = -1 \)); Difference - Different y - intercepts. Observation - Parallel lines (same slope).
- Family D: Similarity - Same y - intercept (\( b = -5 \)); Difference - Different slopes. Observation - Intersect at \( (0, - 5) \) (same y - intercept).
(Graphing: For Family A, plot lines with \( m = 3 \), different \( b \); Family B with \( b = 1 \), different \( m \); etc. Parallel lines (same \( m \)) never meet, lines with same \( b \) meet at y - axis.)