ACT Preparation / Unit 10 ACT 0922 worked example, 60 to 90 seconds
Writing a Transformed Function
The idea
To write a function from a list of transformations, work in three steps. First, start from the base function. Second, apply the changes inside the function, to x, for horizontal moves: replace x with x − h to shift right by h. Third, apply the changes outside, to the output, for vertical moves: multiply by a to stretch or reflect, then add k to shift up. Check one point by tracking it through each move.
In practice
Worked example: take y = √x, reflect it over the x-axis, shift it left 3 and up 2. Inside: x becomes x + 3, giving √(x + 3). Outside: the reflection gives −√(x + 3), and the shift gives −√(x + 3) + 2. Check: the base point (0, 0) moves left to (−3, 0), the reflection leaves it there, and the shift raises it to (−3, 2). At x = −3, −√0 + 2 = 2.
Where people slip
Do the outside steps in order: reflect or stretch first, then shift. Shifting up 2 before reflecting would put the point at −2 instead of 2.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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