Vertical Translation

Basic Concepts of Vertical Translation

Vertical translation is a geometric transformation that shifts the graph of a function up or down along the y-axis without changing the shape of the graph. Imagine lifting or lowering an object vertically, its shape remains the same, only its position changes.

If we have a function $f(x)$, then vertical translation produces a new function $g(x) = f(x) + k$ where $k$ is the translation constant.

Rules of Vertical Translation

For any function $f(x)$, vertical translation is defined as:

Where:

• If $k > 0$, the graph shifts upward by $k$ units
• If $k < 0$, the graph shifts downward by $|k|$ units
• If $k = 0$, there is no translation (graph remains the same)

Visualization of Vertical Translation

Let's see how vertical translation works on the linear function $f(x) = 2x$.

From the visualization above, we can observe:

• The original function $f(x) = 2x$ (purple) passes through the origin
• Function $g(x) = 2x + 3$ (orange) is the result of translation upward by 3 units
• Function $h(x) = 2x + (-2)$ (teal) is the result of translation downward by 2 units

Vertical Translation on Quadratic Functions

Now let's apply the same concept to the quadratic function $f(x) = x^2$.

Notice that:

• The vertex of the original parabola $f(x) = x^2$ is at $(0, 0)$
• After vertical translation +4, the vertex of $g(x) = x^2 + 4$ is at $(0, 4)$
• After vertical translation +(-3), the vertex of $h(x) = x^2 + (-3)$ is at $(0, -3)$

Important Properties of Vertical Translation

Graph Shape Remains Unchanged

Vertical translation preserves the original shape of the graph. The distance between points on the graph remains the same, only the vertical position changes.

Effect on Coordinate Points

If point $(a, b)$ is on the graph of $f(x)$, then after vertical translation by $k$, that point becomes $(a, b + k)$ on the graph of $f(x) + k$.

Domain and Range

• Domain: Does not change after vertical translation
• Range: Shifts by $k$ units

If the range of the original function is $[c, d]$, then the range after vertical translation $k$ becomes $[c + k, d + k]$.

Application Examples

Exponential Function Example

Let's look at vertical translation on the exponential function $f(x) = 2^x$.

For exponential functions:

• The horizontal asymptote $y = 0$ on $f(x) = 2^x$ shifts to $y = k$ on $f(x) + k$
• The y-intercept shifts from $(0, 1)$ to $(0, 1 + k)$

Exercises

1. Given the function $f(x) = x^2 + 4x + 3$. Determine the equation of the function resulting from vertical translation upward by 5 units.

2. If the graph of function $g(x) = 3x + 2$ is translated vertically downward by 7 units, determine:

   • The equation of the resulting translated function
   • The y-intercept after translation

3. Function $h(x) = \sqrt{x}$ undergoes vertical translation such that point $(4, 2)$ becomes $(4, 5)$. Determine the translation constant value and the equation of the resulting translated function.

Answer Key

1. Vertical translation upward by 5 units: $f'(x) = f(x) + 5 = x^2 + 4x + 3 + 5 = x^2 + 4x + 8$

   Function $f(x) = x^2 + 4x + 3$ and Its Translation Result

   Original quadratic function and the result of vertical translation upward by 5 units.

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2. Equation of the resulting translated function:

   • Translation downward by 7 units: $g'(x) = g(x) + (-7) = 3x + 2 + (-7) = 3x + (-5)$
   • Y-intercept: substitute $x = 0$ into $g'(x) = 3(0) + (-5) = -5$, so the intercept point is $(0, -5)$

   Visualization:

   Function $g(x) = 3x + 2$ and Its Translation Result

   Original linear function and the result of vertical translation downward by 7 units.

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3. Point $(4, 2)$ on $h(x) = \sqrt{x}$ becomes $(4, 5)$, meaning vertical translation by $k = 5 - 2 = 3$ units upward. Equation of the translation result: $h'(x) = \sqrt{x} + 3$

   Function $h(x) = \sqrt{x}$ and Its Translation Result

   Original square root function and the result of vertical translation upward by 3 units.

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