Let's analyze the given equations and complete the statements.

(a) Comparing $y = 3x$ and $y = x$:

1. Compared to the graph of $y = x$, the graph of $y = 3x$ is

   • Steeper because the slope (3) is greater than the slope of $y = x$ (which is 1).

2. Compared to the graph of $y = x$, the graph of $y = 3x$ intersects the y-axis at

   • The same point (0,0) because both equations have no y-intercept term, meaning they pass through the origin.

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(b) Comparing $y = x - \frac{4}{3}$ and $y = x$:

1. Compared to the graph of $y = x$, the graph of $y = x - \frac{4}{3}$ is

   • Shifted downward because the $-\frac{4}{3}$ decreases all y-values by $\frac{4}{3}$.

2. Compared to the graph of $y = x$, the graph of $y = x - \frac{4}{3}$ intersects the y-axis at

   • $y = -\frac{4}{3}$ because setting $x = 0$ in $y = x - \frac{4}{3}$ gives $y = -\frac{4}{3}$.

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Would you like a graph to visualize these differences?

Here are five related questions to explore further:

1. What happens to a graph when the slope increases or decreases?
2. How does changing the y-intercept affect the position of a line?
3. What is the significance of parallel lines in the coordinate plane?
4. How do transformations like translations and dilations affect linear functions?
5. How would the graph change if the equation were $y = -3x$ instead of $y = 3x$?

Tip: The slope of a line controls how steep it is, while the y-intercept determines where it crosses the y-axis!