Question

the equation \\(y = 4x\\) represents the relationship between time, \\(x\\), and distance traveled, \\(y\\). which graph represents this relationship?

Explanation:

Analyze the given equation

The given equation is \(y = 4x\). This represents a direct variation or proportional relationship.
Using the Proportional Relationship Equation knowledge point, we know that the graph of a proportional relationship of the form \(y = kx\) must pass through the origin \((0,0)\).

Identify key points on the graph

Using the Constant of Proportionality knowledge point, where \(k = 4\), we can find specific coordinates \((x, y)\) that must lie on the line:

• When \(x = 0\), \(y = 4(0) = 0\). The point is \((0, 0)\).
• When \(x = 1\), \(y = 4(1) = 4\). The point is \((1, 4)\).
• When \(x = 2\), \(y = 4(2) = 8\). The point is \((2, 8)\).
• When \(x = 3\), \(y = 4(3) = 12\). The point is \((3, 12)\).
• When \(x = 4\), \(y = 4(4) = 16\). The point is \((4, 16)\).
• When \(x = 5\), \(y = 4(5) = 20\). The point is \((5, 20)\).

Evaluate the given graph options

Let's check each graph to see which one contains these points:

• First Graph: The line starts at \((0, 40)\) and has a very flat slope. This does not pass through \((0,0)\).
• Second Graph: The line starts at \((0,0)\). Looking at the grid:
• At \(x = 1\), \(y = 4\).
• At \(x = 2\), \(y = 8\).
• At \(x = 3\), \(y = 12\).
• At \(x = 4\), \(y = 16\).
• At \(x = 5\), \(y = 20\).

This perfectly matches our calculated points.

• Third Graph: The line starts at \((0, 4)\) and goes to \((5, 9)\). This does not pass through \((0,0)\).
• Fourth Graph: This graph is blank/incomplete.

Therefore, the second graph is the correct representation.

Answer:

• (A) Graph starting at (0, 40) with a very flat slope
• (B) Graph starting at (0, 0) and passing through (5, 20) (Correct answer)
• (C) Graph starting at (0, 4) and passing through (5, 9)
• (D) Blank graph grid