Linear Function y=mx+b: Converting between graphs and equations

To solve this problem, we'll follow these steps:

• Determine if the given line labeled '1' has a positive or negative slope.
• Find out if the line passes through the origin, which implies the y-intercept, $b = 0$.
• Identify the function equation based on the observations.

Now, let's work through each step:
- Observation shows that line '1' in red goes through the origin (intersects at the point (0,0)), confirming that the y-intercept, $b$, is zero.
- Given that this line descends from left to right, the slope $m$ is negative. Therefore, the most likely function representation given the described visual descent is $y = -x$.

Thus, the function describing the line numbered '1' is $y = -x$.

To solve this problem, we need to determine the function that corresponds to the line labeled "4" on the number line.

The line labeled "4" is depicted in orange and appears to pass through the origin, indicating that the y-intercept is 0. Therefore, the equation has the form $y = mx$.

Upon examining this line, we observe that it is direct and passes through the origin at a $45^\circ$ angle to both axes, suggesting that the slope $m$ is 1. This is to say, for each unit increase in $x$, $y$ also increases by the same magnitude.

• The equation thus simplifies to $y = x$.

We next compare our result to the given function choices:

• Choice A: $y = -x$ (Negative slope)
• Choice B: $y = x$ (Positive slope of 1)
• Choice C: $y = 6x$ (Steeper positive slope)
• Choice D: $y = -6x$ (Steeper negative slope)

The line labeled "4" directly matches Choice B, $y = x$, as both the direction and slope align with our observations.

Therefore, the function corresponding to the number line labeled "4" is $y = x$.