Question

make a connection consider the graph of the equation $y = mx + b$.
a. how does changing the value of $m$ affect the graph of the equation?
b. how does changing the value of $b$ affect the graph of the equation?

Explanation:

a. In the linear equation $y=mx+b$, $m$ represents the slope of the line. Changing $m$ alters the steepness and direction of the line: positive $m$ makes the line rise from left to right, negative $m$ makes it fall, larger absolute values of $m$ make the line steeper, and $m=0$ gives a horizontal line.
b. In the linear equation $y=mx+b$, $b$ represents the y-intercept (the point where the line crosses the y-axis). Changing $b$ shifts the entire line vertically: increasing $b$ shifts the line upward, decreasing $b$ shifts the line downward, without changing the line's slope.

Answer:

a. Changing $m$ modifies the line's slope (steepness and direction): larger $|m|$ = steeper line; positive $m$ rises left-to-right, negative $m$ falls left-to-right; $m=0$ gives a horizontal line.
b. Changing $b$ shifts the line vertically: higher $b$ shifts the line up, lower $b$ shifts the line down, with no change to the line's slope.