Horizontal Dilation

Basic Concepts of Horizontal Dilation

Horizontal dilation is a geometric transformation that changes the size of a function graph horizontally, like pulling or compressing a rubber band left and right. Imagine holding a photo with both hands on the left and right sides, then stretching or compressing it horizontally without changing the height of the photo.

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

Rules of Horizontal Dilation

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

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

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

Note that the effect of horizontal dilation is counterintuitive: a larger scale factor actually compresses the graph.

Visualization of Horizontal Dilation

Let's see how horizontal 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 horizontally compressed by factor 2
• Function $h(x) = (0.5x)^2$ (sky) is horizontally stretched by factor 0.5
• All graphs have the same vertex at $(0, 0)$

Horizontal Dilation on Linear Functions

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

Notice that:

• The original function $f(x) = x + 2$ has slope 1
• Function $g(x) = f(2x) = 2x + 2$ has slope 2 (horizontally compressed)
• Function $h(x) = f(0.5x) = 0.5x + 2$ has slope 0.5 (horizontally stretched)
• All lines intersect the y-axis at the same point $(0, 2)$

Important Properties of Horizontal Dilation

Effect on Coordinate Points

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

Domain and Range

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

If the domain of the original function is $[c, d]$, then the domain after horizontal dilation with factor $k > 0$ becomes $[\frac{c}{k}, \frac{d}{k}]$.

Axis Intercepts

• x-intercept: Changes according to the scale factor
• y-intercept: Does not change

Application Examples

Exponential Function

Let's look at horizontal 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 remains the same at $(0, 1)$
• The growth rate of the function changes according to the scale factor

Horizontal 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 y-axis
• Simultaneously undergoes dilation according to the absolute value of the scale factor
• The graph shape remains the same because the quadratic function is symmetric

Exercises

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

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

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

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

Answer Key

1. Horizontal dilation with factor 3: $f'(x) = f(3x) = (3x)^2 - 2(3x) + 1 = 9x^2 - 6x + 1$

   Function $f(x) = x^2 - 2x + 1$ and Its Dilation Result

   The original parabola is horizontally compressed by factor 3 producing a narrower parabola.

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2. Equation of the resulting dilated function:

   • Horizontal dilation: $g'(x) = g(\frac{1}{2}x) = \sqrt{\frac{1}{2}x} = \sqrt{\frac{x}{2}}$
   • Domain after dilation: $[0, \infty)$ becomes $[0, \infty)$ (unchanged because the scale factor is positive)

   Visualization:

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

   The square root curve is horizontally stretched by factor 0.5 producing a wider curve.

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3. The original function $h(x) = |x - 2|$ has its vertex at $(2, 0)$. After horizontal dilation with factor -1: $h'(x) = |-x - 2| = |-(x + 2)| = |x + 2|$, the vertex becomes $(-2, 0)$.

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

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

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