Vertical Shifts of Linear Equations

Property The graph of an equation in the form $y = mx + b$ is a vertical translation (or shift) of the graph of $y = mx$. The value of $b$ determines the direction and magnitude of the shift. If $b 0$, the graph of $y = mx$ is shifted up by $b$ units. If $b < 0$, the graph of $y = mx$ is shifted down by $|b|$ units.

Examples The graph of $y = 2x + 5$ is the graph of $y = 2x$ shifted vertically upwards by 5 units. The graph of $y = x 3$ is the graph of $y = x$ shifted vertically downwards by 3 units. The graph of $y = \frac{1}{2}x + 1$ is the graph of $y = \frac{1}{2}x$ shifted vertically upwards by 1 unit.

Explanation This skill provides a way to visualize graphing two step equations by transforming a simpler one step equation. Since the line $y = mx$ always passes through the origin $(0, 0)$, the constant term $b$ shifts the y intercept from $(0, 0)$ to $(0, b)$. For every x value, the corresponding y value on the line $y = mx + b$ is simply $b$ more than the y value on the line $y = mx$. This results in the entire line moving up or down the y axis without changing its steepness (slope).