Does a vertical stretch move the vertical asymptote?

No. The vertical stretch factor A does not appear in the vertical asymptote formula x = a/B + h. Only the horizontal stretch B and the horizontal shift h change the vertical asymptote's location.

What happens to a vertical asymptote when B is negative?

A negative B reflects the graph across the y-axis. The vertical asymptote x = a becomes x = a/B. If a is positive, the asymptote moves to a negative x-value. If a is zero, it stays at zero.

Can a graph touch its vertical asymptote after a transformation?

No. A vertical asymptote is a line where the function is unbounded. No transformation can make the function defined at that x-value if the factor causing the asymptote does not cancel. The graph approaches the asymptote but never touches it.

If a horizontal asymptote is at y = 0 and I shift up by 3, does the graph still have end behavior at y = 0?

No. The horizontal asymptote moves to y = 3. The graph's end behavior approaches y = 3, not y = 0. The shift applies to the asymptote just as it applies to every point.

What happens to a hole when the function is transformed?

A hole moves with the same transformation rules as a vertical asymptote. The x-coordinate of the hole shifts by h and is divided by B. The y-coordinate of the hole is multiplied by A and shifted by k. The hole remains a removable discontinuity.

Does a reflection across the x-axis change a slant asymptote?

Yes. For a slant asymptote y = mx + b, the reflection multiplies the slope and intercept by A = −1. The new slant asymptote is y = −mx − b after the vertical shift k is added. The horizontal shift h does not affect the slope, only the x-intercept of the line.

Why does my calculator show a vertical asymptote where there is a hole?

The calculator is not simplifying the function before checking for asymptotes. For f(x) = (x² − 1)/(x − 1), the factor (x − 1) cancels. The calculator sees the denominator root at x = 1 and reports a vertical asymptote. The correct classification is a hole. Always simplify the function manually before trusting a calculator's asymptote output.

How Function Transformations Move Asymptotes and Restrictions

You graph f(x) = 1/x. You see the vertical asymptote at x = 0, the horizontal asymptote at y = 0. Now you shift the graph right by 4, stretch it vertically by 2, and lift it by 5. The asymptotes move with the graph, but the movement follows a predictable algebra, not guesswork. This is how function transformations move asymptotes and restrictions: each transformation type, shift, stretch, reflection, applies to the asymptote line exactly as it applies to every point on the graph.

Start with the formula. For a base function f(x) and a transformed function g(x) = A · f(B(x − h)) + k, the vertical asymptote at x = a shifts to x = a/B + h. The horizontal asymptote at y = b becomes y = A · b + k. The same rule applies to domain restrictions (x-values the function cannot take) and range restrictions (y-values the function cannot reach). Track the transformed domain and range alongside each moved asymptote to avoid errors later.

A Worked Example

Work one concrete example from start to finish. Use f(x) = (x − 1)/(x + 2). The vertical asymptote is at x = −2, found by setting the denominator to zero. The horizontal asymptote is at y = 1, because the degrees of numerator and denominator are equal and the ratio of leading coefficients is 1/1.

Apply the transformation g(x) = −f(x − 3) + 2. The horizontal shift is h = 3 to the right. The vertical stretch factor is A = −1 (a reflection across the x-axis). The vertical shift is k = 2 upward.

For the vertical asymptote: take the original x = −2. Divide by B = 1 (no horizontal stretch or reflection), then add h = 3. The vertical asymptote moves to x = 1. For the horizontal asymptote: take the original y = 1. Multiply by A = −1, then add k = 2. The horizontal asymptote moves to y = 1. That is not a typo: −1 + 2 = 1. The graph is reflected and shifted, and the asymptote ends up at the same y-value it started at, but for a different reason.

Now track the domain. The original domain excludes x = −2. After the transformation, the excluded x-value shifts to x = 1. The range of f(x) is all real numbers except y = 1. After the transformation, the range still excludes y = 1. The reflection and shift cancelled each other out.

To verify, use a domain and range calculator. Enter f(x) and g(x) separately and compare the asymptote outputs. If the calculator does not simplify the function first, it may report a vertical asymptote at the hole's x-value, a known failure mode for calculators that skip the simplification step.

Calculation Steps

Apply the calculation rules step by step to any rational function. The formula g(x) = A · f(B(x − h)) + k has four parameters that affect asymptotes.

Vertical Asymptote: x = a → x = a/B + h

The horizontal stretch factor B divides the original x-coordinate of the vertical asymptote. If B = 3, the asymptote moves to one-third of its original distance from zero before the horizontal shift is added. If B = −1, the asymptote reflects across the y-axis. For f(x) = 1/x with a vertical asymptote at x = 0, the reflection leaves it unchanged: 0/(-1) = 0.

Horizontal Asymptote: y = b → y = A · b + k

The vertical stretch factor A multiplies the original y-value of the horizontal asymptote. Then k shifts it. If A is negative, the asymptote reflects across the x-axis. For f(x) = 1/x with y = 0, the multiplication does nothing because 0 times anything is 0. Only the vertical shift k moves that asymptote.

Domain and Range Restrictions

Every excluded x-value from the original domain moves with the same formula as vertical asymptotes. Every excluded y-value from the original range moves with the same formula as horizontal asymptotes. If the original function has a hole at x = 3 with y = 2, the transformed hole appears at x = 3/B + h and y = A · 2 + k.

Why the Method Works

The explanation reduces to one idea: transformations act on the entire coordinate system of the graph, including the asymptote lines. Vertical shifts move horizontal asymptotes. Horizontal shifts move vertical asymptotes. Vertical stretches multiply the horizontal asymptote value. Horizontal stretches divide the vertical asymptote x-coordinate. Reflections change the sign of the asymptote's position or value.

The common mistake is treating asymptotes as fixed barriers. A horizontal asymptote defines end behavior only. The graph can cross a horizontal asymptote at finite x-values. A vertical asymptote, by contrast, is a line the graph never touches because the function is unbounded there. After a transformation, the graph still cannot touch its vertical asymptote, but it can cross its horizontal asymptote.

Track the original function's end behavior. If f(x) approaches y = b as x → ±∞, then g(x) approaches y = A · b + k as x → ±∞. This is the same limit rule applied to the transformed function. The horizontal stretch B does not affect end behavior in the y-direction, but it does affect how quickly the graph approaches the asymptote.