Question

investigating the y-intercept
try it
use the slider to change the value of b. which statement is true?

• if the value of b is increased from 0, the graph moves up.
• if the value of b is increased from 0, the graph gets less steep.
• if the value of b is decreased from 0, the graph moves up.
• if the value of b is decreased from 0, the graph gets steeper.

(graph: y = x + b, b = 0, coordinate grid with the line passing through the origin)

Explanation:

Step1: Recall Slope-Intercept Form

The equation of the line is \( y = x + b \), which is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. Here, the slope \( m = 1 \) (constant) and \( b \) is the y - intercept.

Step2: Analyze the effect of changing \( b \)

• For the slope: Since the slope \( m = 1 \) (coefficient of \( x \)) does not depend on \( b \), changing the value of \( b \) will not change the steepness of the line (steepness is determined by the slope). So options about the steepness changing (the second and fourth options) are incorrect.
• For the y - intercept: The y - intercept is the point where the line crosses the y - axis (\( x = 0 \)). If \( b \) increases from 0, the y - intercept \( (0,b) \) moves up along the y - axis, so the graph of the line moves up. If \( b \) decreases from 0, the y - intercept \( (0,b) \) moves down along the y - axis, so the graph of the line moves down. So the third option is incorrect and the first option is correct.

Answer:

If the value of \( b \) is increased from 0, the graph moves up.