Transformations Of Rational Functions

Find oblique asymptotes and analyze end behavior of rational functions. Math 30-1.

Lesson 6.5 of Radical & Rational Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction — What Is a Rational Function?. Discover what rational functions are, explore the parent function y = 1/x, and understand NPVs, asymptotes, domain, and range.
• Transformations & Reading the Graph. Master how each parameter transforms y = 1/x, analyze complete examples, and learn to read key features directly from the equation.
• Synthetic Division — Converting to Standard Form. Learn how to convert disguised rational functions into standard form using synthetic division and discover the time-saving shortcut.
• Writing Equations. Master the technique of finding the equation of a rational function when given a graph with asymptotes and a point.

Inside the lesson — a free preview

What Is a Rational Function?

The Big Idea. A rational function is simply a fraction where both the numerator and the denominator are polynomials.

Rational Function = (Polynomial)/(Polynomial)

These functions are powerful because they can model situations involving rates, ratios, and proportions—think of them as "division functions"!

Examples of Rational Functions. y = (1)/(x) (parent function)

y = (2x+5)/(x-3) (linear over linear)

y = (x^2+4)/(x^2-9) (quadratic over quadratic)

The Parent Function: y = (1)/(x)

The Simplest Rational Function. Just like y = x^2 is the parent quadratic function, the simplest rational function is:

y = (1)/(x)

Let's build a table of values to understand its behavior:

Key Observation. The graph has two separate branches:

• Quadrant I (top-right): Both x and y are positive

• Quadrant III (bottom-left): Both x and y are negative

The two branches never touch and never cross the axes.

Non-Permissible Values (NPVs)

Why Can't We Divide by Zero?. When x = 0, we try to calculate:

y = (1)/(0)

This is undefined! We call these values Non-Permissible Values (NPVs).

Finding NPVs — The Rule. To find NPVs, set the denominator equal to zero and solve:

x - h = 0 → x = h

Example: For y = (2)/(x-5):

x - 5 = 0 → x = 5

The NPV is x = 5 because that makes the denominator zero.

Asymptotes — Lines We Approach But Never Touch

What Is an Asymptote?. An asymptote is an imaginary line that the graph gets closer and closer to but never actually reaches.

Asymptotes for y = (1)/(x). Vertical Asymptote (VA): x = 0 (the y-axis)

• The graph approaches but never touches the y-axis

• As x → 0, the function values get infinitely large

Horizontal Asymptote (HA): y = 0 (the x-axis)

• The graph approaches but never touches the x-axis

• As x → ±∞, the function values approach 0

Memory Trick. 🎯 The Vertical Asymptote happens at the NPV (where the denominator is zero).

📈 The Horizontal Asymptote shows where the graph settles as x → ±∞.

Worked example: Finding g(x) from a Description of Transformations

The Problem. Given: Start with f(x) = (1)/(x). Apply: Vertical stretch by 4, Horizontal stretch by (1)/(2), reflect in x-axis, translate 3 right and 5 down.

Step 1 — Find a. We need to apply three transformations that affect a:

1. Vertical Stretch by 4: This multiplies the function by 4, so a starts at 4.

2. Horizontal Stretch by 1/2: For y = (1)/(x), stretching horizontally by (1)/(2) means replacing x with 2x. This gives y = (1)/(2x), which is the same as multiplying by (1)/(2).

Combined effect: a = 4 × (1)/(2) = 2

3. Reflection in x-axis: This makes a negative.

Final answer: a = -2

Step 2 — Find h and k. Translate 3 right: h = 3

Translate 5 down: k = -5

Answer. Function: g(x) = (-2)/(x-3) - 5

Asymptotes: VA: x = 3 (denominator = 0 when x = 3) | HA: y = -5 (k value)

Domain: (-∞, 3) (3, ∞)

Range: (-∞, -5) (-5, ∞)

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