Question

(b) complete the statements below about the graphs of y = x - 3 and y = x. compared to the graph of y = x, the graph of y = x - 3 is select. compared to the graph of y = x, the graph of y = x - 3 intersects the y - axis at select.

Explanation:

To solve this, we analyze the transformation of the linear function \( y = x \) to \( y = x - 3 \) and their y - intercepts.

Step 1: Analyze the vertical shift of the graph

The general form of a linear function is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For the function \( y = x \), we can write it as \( y=1\times x+0 \), so the slope \( m = 1 \) and the y - intercept \( b = 0 \). For the function \( y=x - 3 \), we can write it as \( y = 1\times x+( - 3) \), so the slope \( m = 1 \) and the y - intercept \( b=-3 \).

When the slope of two linear functions is the same, the graphs are parallel. Since both \( y = x \) and \( y=x - 3 \) have a slope of \( 1 \), the graph of \( y=x - 3 \) is a vertical shift (downward shift of 3 units) of the graph of \( y = x \), and the two lines are parallel. So the first blank should be filled with "a vertical shift down 3 units (or parallel and shifted down 3 units)".

Step 2: Analyze the y - intercept

The y - intercept of a function is the point where the graph intersects the y - axis. To find the y - intercept, we set \( x = 0 \) in the function.

• For \( y=x \), when \( x = 0 \), \( y=0 \). So the y - intercept of \( y = x \) is \( (0,0) \).
• For \( y=x - 3 \), when \( x = 0 \), \( y=0 - 3=-3 \). So the y - intercept of \( y=x - 3 \) is \( (0,-3) \).

Compared to the y - intercept of \( y = x \) (which is at \( y = 0 \)), the y - intercept of \( y=x - 3 \) is at \( y=-3 \), so the graph of \( y=x - 3 \) intersects the y - axis at a lower point (or at \( (0, - 3) \) instead of \( (0,0) \)).

For the first statement:

The functions \( y = x \) and \( y=x - 3 \) have the same slope (\( m = 1 \)). When the slope of two linear functions is equal, the graphs are parallel, and \( y=x - 3 \) is obtained by shifting \( y = x \) down 3 units.

The y - intercept of \( y = x \) is \( (0,0) \) (when \( x = 0,y = 0 \)), and the y - intercept of \( y=x - 3 \) is \( (0,-3) \) (when \( x = 0,y=-3 \)). So compared to \( y = x \), \( y=x - 3 \) intersects the y - axis at a lower y - value (at \( y=-3 \) instead of \( y = 0 \)).

Answer:

a vertical shift down 3 units (or parallel and shifted down 3 units)

For the second statement: