Which diagram shows the equation
? ? View Available Hint(s)
The Correct Answer and Explanation is:
ChatGPT said:
The given equation is:v=2t+4v = 2t + 4v=2t+4
This is a linear equation where:
- vvv is the dependent variable (on the y-axis),
- ttt is the independent variable (on the x-axis),
- The slope (rate of change) is 2,
- The y-intercept is 4 (meaning when t=0t = 0t=0, v=4v = 4v=4).
Step-by-step Analysis:
Let’s analyze each graph in the image:
Top graph:
- The line starts at v=2v = 2v=2 when t=0t = 0t=0.
- This suggests a y-intercept of 2.
- Since our equation has a y-intercept of 4, this is not correct.
Middle graph:
- The line starts at v=4v = 4v=4 when t=0t = 0t=0.
- Let’s test a point: When t=1t = 1t=1, v=6v = 6v=6, and when t=2t = 2t=2, v=8v = 8v=8. This shows a consistent increase of 2 units in vvv for every 1 unit in ttt.
- This confirms a slope of 2 and y-intercept of 4.
✅ This matches the equation v=2t+4v = 2t + 4v=2t+4. Correct Answer: Middle graph
Bottom graph:
- The line starts at v=6v = 6v=6 when t=0t = 0t=0, so y-intercept is 6.
- That doesn’t match the y-intercept of 4.
- Also, the slope appears steeper than 2 (looks like slope = 3).
This graph is incorrect for our equation.
Explanation
To determine which graph correctly represents the equation v=2t+4v = 2t + 4v=2t+4, we analyze the graph’s slope and y-intercept. A linear equation in the form y=mx+by = mx + by=mx+b has two key characteristics: the slope (rate of change) and the y-intercept (value when x=0x = 0x=0). For the equation v=2t+4v = 2t + 4v=2t+4, the slope is 2 and the y-intercept is 4. This means that when t=0t = 0t=0, vvv must equal 4. Additionally, for every increase of 1 in ttt, vvv must increase by 2.
Reviewing the three graphs, the top graph has a line starting at v=2v = 2v=2, which does not match the y-intercept. The bottom graph starts at v=6v = 6v=6, again not matching the required y-intercept of 4. Only the middle graph correctly shows a line that begins at v=4v = 4v=4 and increases by 2 for every 1 unit increase in ttt. This accurately reflects the slope and intercept in the equation.
Hence, the middle graph is the correct visual representation of the equation v=2t+4v = 2t + 4v=2t+4.
