Vertical Dilation

Basic Concepts of Vertical Dilation

Vertical dilation is a geometric transformation that changes the size of a function graph vertically, like stretching or compressing a rubber band up and down. Imagine pulling a photo with both hands, one above and one below, then stretching or compressing it vertically without changing the width of the photo.

If we have a function $f(x)$, then vertical dilation produces a new function $g(x) = k \cdot f(x)$ where $k$ is the scale factor that determines how much the vertical size changes.

Rules of Vertical Dilation

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

Where $k$ is the scale factor that affects the transformation:

• If $k > 1$, the graph is stretched vertically (enlarged)
• If $0 < k < 1$, the graph is compressed vertically (reduced)
• If $k = 1$, the graph does not change
• If $k < 0$, the graph undergoes reflection as well as dilation

Visualization of Vertical Dilation

Let's see how vertical dilation works on the quadratic function $f(x) = x^2$ with various scale factors.

From the visualization above, we can observe:

• The original function $f(x) = x^2$ (purple) as reference
• Function $g(x) = 2x^2$ (orange) is vertically stretched by factor 2
• Function $h(x) = 0.5x^2$ (sky) is vertically compressed by factor 0.5
• All graphs have the same vertex at $(0, 0)$

Vertical Dilation on Linear Functions

Now let's apply the same concept to the linear function $f(x) = x + 1$.

Notice that:

• The original function $f(x) = x + 1$ has slope 1
• Function $g(x) = 3(x + 1)$ has slope 3 (stretched)
• Function $h(x) = 0.5(x + 1)$ has slope 0.5 (compressed)
• All lines still intersect the y-axis, but at different points

Important Properties of Vertical Dilation

Effect on Coordinate Points

If point $(a, b)$ is on the graph of $f(x)$, then the corresponding point on the graph of $k \cdot f(x)$ is $(a, k \cdot b)$.

Domain and Range

• Domain: Does not change after vertical dilation
• Range: Changes according to the scale factor $k$

If the range of the original function is $[c, d]$, then the range after vertical dilation with factor $k > 0$ becomes $[k \cdot c, k \cdot d]$.

Axis Intercepts

• x-intercept: Does not change (except if $k = 0$)
• y-intercept: Changes according to the scale factor

Application Examples

Exponential Function

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

For exponential functions:

• The horizontal asymptote remains at $y = 0$ for both functions
• The y-intercept changes from $(0, 1)$ to $(0, 2)$
• The growth rate of the function increases according to the scale factor

Vertical Dilation with Negative Factor

Let's see what happens when the scale factor is negative.

When the scale factor is negative:

• The graph undergoes reflection across the x-axis
• Simultaneously undergoes dilation according to the absolute value of the scale factor
• A parabola that opens upward becomes one that opens downward

Exercises

1. Given the function $f(x) = x^2 - 4x + 3$. Determine the equation of the function resulting from vertical dilation with scale factor 3.

2. If the graph of function $g(x) = \sqrt{x}$ undergoes vertical dilation with factor $\frac{1}{2}$, determine:

   • The equation of the resulting dilated function
   • The range of the function after dilation

3. Function $h(x) = |x - 1| + 2$ undergoes vertical dilation with factor -1. Determine the vertex of the resulting dilated function.

Answer Key

1. Vertical dilation with factor 3: $f'(x) = 3f(x) = 3(x^2 - 4x + 3) = 3x^2 - 12x + 9$

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

   The original parabola is vertically stretched by factor 3 producing a taller parabola.

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2. Equation of the resulting dilated function:

   • Vertical dilation: $g'(x) = \frac{1}{2}g(x) = \frac{1}{2}\sqrt{x}$
   • Range after dilation: $[0, \infty)$ becomes $[0, \infty)$ but with smaller maximum values

   Visualization:

   Function $g(x) = \sqrt{x}$ and Its Dilation Result

   The square root curve is vertically compressed by factor 0.5 producing a lower curve.

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3. The original function $h(x) = |x - 1| + 2$ has its vertex at $(1, 2)$. After vertical dilation with factor -1: $h'(x) = -1 \cdot (|x - 1| + 2) = -|x - 1| - 2$, the vertex becomes $(1, -2)$.

   Function $h(x) = |x - 1| + 2$ and Its Dilation Result

   The absolute value function undergoes reflection and dilation with factor -1.

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