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8th Grade Math: Graph Andre's Distance as a Function of Time for Football Motion
Mathematics
Grade 8 (Junior High School)
Question Content
Draw a graph of Andre's distance as a function of time for this situation: When the football play started, Andre ran forward 20 yards, then turned around and ran 5 yards back. He stood in that spot for 3 seconds, then walked back to where he began. Label the axes appropriately. You do not have to include numbers on the axes or the coordinates of points on your graph.
Correct Answer
A graph with: 1. Horizontal axis labeled "Time", vertical axis labeled "Distance from Starting Point"; 2. First segment: steep upward sloping line (forward 20 yards over short time); 3. Second segment: downward sloping line (back 5 yards over short time); 4. Third segment: horizontal line (stationary for 3 seconds); 5. Fourth segment: gentle downward sloping line (slowly walking back to starting point over longer time).
Detailed Solution Steps
1
Step 1: Label the axes. The horizontal axis represents the independent variable, which is "Time", and the vertical axis represents the dependent variable, which is "Distance from Starting Point".
2
Step 2: Graph the first action: Andre runs forward 20 yards quickly. This is a steep upward-sloping line, as distance from the start increases rapidly over a short time.
3
Step 3: Graph the second action: Andre runs 5 yards back quickly. This is a downward-sloping line, as his distance from the start decreases over a short time.
4
Step 4: Graph the third action: Andre stands still for 3 seconds. Since his distance does not change, this is a horizontal flat line that stays at the same vertical level for a noticeable length of the time axis.
5
Step 5: Graph the fourth action: Andre walks back to the starting point slowly. This is a gentle downward-sloping line (less steep than the running back segment) that goes from his current distance down to 0 on the vertical axis, over a longer time interval than the running segments.
Knowledge Points Involved
1
Distance-Time Graphs
A distance-time graph plots distance from a starting point (vertical axis) against time (horizontal axis). The slope of the line represents speed: steeper slopes mean faster speed, horizontal lines mean no movement, and upward/downward slopes mean movement away from/toward the starting point.
2
Independent and Dependent Variables
In a function, the independent variable (time, in this case) is the input that is controlled or measured, and the dependent variable (distance, in this case) is the output that depends on the input. On a graph, the independent variable is on the horizontal axis, and the dependent variable is on the vertical axis.
3
Graphing Real-World Motion
When graphing motion, different types of movement correspond to specific line segments on a distance-time graph: constant speed (straight line), stationary (horizontal line), and changing speed (curved line, though this problem only involves constant speed segments).
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