Translate functions and write equations of shifted graphs. Grade 12 Math 30-1 Alberta.
Lesson 1.6 of Function Transformations & Operations in Math 30-1: Alberta curriculum lessons.
A translation is a transformation that shifts a graph up, down, left, or right without changing its shape. The graph looks exactly the same, it just moves to a new position.
Key Characteristics.
Before we translate anything, we need a starting point called the parent function.
Definition. The parent function is the simplest form of a function, written as y = f(x). It has no transformations applied, no shifts, stretches, or reflections.
Examples of Parent Functions.
These are the basic, untransformed versions. All transformations start from these parent functions.
When we translate a function, we write it in this form:
y = f(x - h) + k
This looks complicated, but it's just telling us how to move the graph:
What Each Parameter Does.
💡 Remember. Both h and k are just numbers! They tell us how many units to move in each direction.
The value h shifts the graph horizontally (left or right).
The Rules for h.
⚠️ Important: The Direction is Opposite!. Notice that the direction is opposite of what you might expect. When we see f(x - 3), the graph moves right, not left!
We'll explore why this happens later in this lesson.
Examples.
The value k shifts the graph vertically (up or down).
The Rules for k.
✅ Good News!. This one works exactly how you'd expect! Positive k moves up, negative k moves down. No surprises here.
Examples.
y = f(x - h) + k
Remember These Points.
A horizontal translation shifts a graph left or right without changing its shape.
The general form is:
y = f(x - h)
where h controls how far left or right the graph moves.
Given. Parent function: f(x) = x^2
Translated function: y = (x - 4)^2
Analysis. • h = 4
• Since h > 0, the graph moves right 4 units
See the graph below showing the black parabola (original at x = 0) and the red parabola (shifted right to x = 4).
Result.
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