Math Problem Statement
Two lines are graphed on the coordinate plane.
Which description of the lines is TRUE?
A. The lines have the same slopes, but different -intercepts. The lines are a translation of each other by units.
B. The lines have the same slopes, but different -intercepts. The lines are a translation of each other by units.
C. The lines have different slopes, but the same -intercepts. The lines are a translation of each other by units.
D. The lines have different slopes, but the same -intercepts. The lines are a translation of each other by units.
Solution
To determine the correct answer, let's analyze the properties of the two lines based on their slopes and -intercepts.
-
Same Slopes or Different Slopes:
If the lines are parallel, they will have the same slope. If they are not parallel (meaning they intersect or are angled differently), they will have different slopes. -
Same -Intercepts or Different -Intercepts:
If the lines cross the -axis at the same point, they have the same -intercept. Otherwise, they have different -intercepts. -
Translation
Translating one line to overlap with another typically involves shifting vertically, horizontally, or both, depending on the direction and distance. This shift is often specified in units along the - or -axis.
From the options:
- Option A suggests the lines are parallel (same slope) but have different -intercepts.
- Option B also suggests parallel lines with different -intercepts but may imply a specific unit shift.
- Options C and D indicate the lines have different slopes, which would mean they intersect or are angled differently.
To determine the best choice, consider that:
- If the lines appear to be parallel, A or B would be correct.
- If they appear to intersect or are angled, C or D would be correct.
Without viewing the exact graph here, a parallel interpretation aligns with options A and B.
If the graph clearly shows a parallel relationship, the correct answer would likely be B if the specific unit shift is confirmed in the context.
If you'd like, upload the graph image for a more precise determination!
Would you like further explanation on translations or more examples?
Here are five additional questions for practice:
- How do you determine the slope of a line from a graph?
- What is a -intercept, and why is it significant in graphing lines?
- What transformations exist beyond translations that can map one line onto another?
- How do parallel lines differ from perpendicular lines in terms of slopes?
- Can two lines with the same slope and different -intercepts ever intersect?
Tip: When identifying transformations on a graph, first check if the slopes match for a quick determination of whether lines are parallel!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Parallel Lines
Translations
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Translation transformation formula
Theorems
Properties of parallel lines
Properties of translations in coordinate geometry
Suitable Grade Level
Grades 8-10
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