Cogito
AP Precalculus · Chapter 4 · Lesson 1
Transformations of Functions
Inside the bracket goes the wrong way.
11 problems · about 22 minutes · AP Precalculus 1.13, 1.14
What this lesson teaches
The student predicts the effect of each transformation on a graph and applies several in order.
- Outside the function acts vertically and behaves as written.
- Inside acts horizontally and runs backwards.
- Several transformations together are order-dependent.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x + 9) moves the graph left by how much?
Answer 9
Why Left 9.
f(x − 7). Right by how much?
Answer 7
Why Minus inside means right.
f(x) + 2. Up by how much?
Answer 2
Why Outside as written.
3f(x). Vertical stretch factor?
Answer 3
Why The multiplier outside.
Build It Up
The same ideas with more to keep track of.
3 problemsMatch each transformation to its effect.
Answer f(x) + k → Shift up by k; f(x + k) → Shift left by k; −f(x) → Reflect across the x-axis; f(−x) → Reflect across the y-axis
Why Inside is horizontal; outside is vertical.
f(2x). Horizontal compression factor?
Answer 2
Why Inside reverses, so it compresses.
Order the steps for graphing y = 2f(x − 3) + 1 from the base graph.
Answer 1. Shift right 3 2. Stretch vertically by 2 3. Shift up 1
Why Horizontal first, then the vertical stretch, then the vertical shift.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsThe point (2, 5) is on y = f(x). What is its y on y = f(x) + 4?
Answer 9
Why Only the output moves.
The point (2, 5) is on y = f(x). What is its x on y = f(x − 3)?
Answer 5
Why The graph moves right 3.
Inside the Bracket: f(x + 4) moves the graph which way? Answer 1 for left, 2 for right.
Answer 1
Why Left 4.
Outside the Bracket: f(x) − 6 moves the graph down by how much?
Answer 6
Why Down 6.