Question

Graph the following linear equations on the grid. The equation y = x has been graphed for you. For

each new equation explain what the number 3 does to the graph of y = x. Pay attention to the
y-intercept, the x-intercept, and the slope. Identify what changes in the graph and what stays the
same.
a. y = x + 3
b. y = x - 3
C. y = 3x"

Answer 1

a. The equation y = x + 3 shifts the graph of y = x upward by 3 units without changing its slope. It retains the same slope of 1 but has a y-intercept of (0, 3) instead of (0, 0).

b. The equation y = x - 3 shifts the graph of y = x downward by 3 units without altering its slope. It maintains the same slope of 1 but now has a y-intercept of (0, -3) rather than (0, 0).

c. The equation y = 3x alters the graph of y = x by changing its slope to 3. It increases the steepness of the line, modifying the slope from 1 to 3 while the y-intercept remains at (0, 0).

Step-by-step explanation:

a. In the equation y = x + 3, the "+3" term translates the graph of y = x vertically upward by three units. This addition of a constant term does not change the slope, which remains 1. However, it shifts the y-intercept from (0, 0) to (0, 3), indicating that the line now intersects the y-axis at y = 3 while retaining the same slope.

b. For y = x - 3, the "-3" term shifts the graph of y = x vertically downward by three units while maintaining the slope of 1. This adjustment in the equation alters the y-intercept from (0, 0) to (0, -3), signifying that the line now crosses the y-axis at y = -3 while keeping the same slope.

c. In the equation y = 3x, the coefficient "3" modifies the slope of the graph from 1 (as in y = x) to 3. This change increases the steepness of the line without altering its y-intercept, which remains at (0, 0). Consequently, the line becomes steeper, with a rise of 3 units for every run of 1 unit along the x-axis, indicating the change in slope.