M2L6 Transformations of Functions
- 2. Parent functions are the most basic form of a function. They are centered or oriented around the origin (0,0). See some of our most common parent functions below: 1. Linear 𝒚 = 𝒙 2. Quadratic 𝒚 = 𝒙 𝟐 3. Cubic 𝒚 = 𝒙 𝟑 4. Radical 𝒚 = 𝒙 5. Absolute value 𝒚 = 𝒙 6. Exponential 𝒚 = 𝟐 𝒙 7. Rational 𝒚 = 𝟏 𝒙
- 3. Putting numbers into the parent function transforms it into something new. The letters a, h, & k are used to represent where we place numbers. 1. Linear 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌 2. Quadratic 𝒚 = 𝒂(𝒙 − 𝒉) 𝟐 + 𝒌 3. Cubic 𝒚 = 𝒂(𝒙 − 𝒉) 𝟑 + 𝒌 4. Radical 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌 5. Absolute value 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌 6. Exponential 𝒚 = 𝒂(𝟐 𝒙−𝒉 ) + 𝒌 7. Rational 𝒚 = 𝒂 𝟏 𝒙−𝒉 + 𝒌 or 𝒚 = 𝒂 𝒙−𝒉 + 𝒌
- 4. First, let’s look at what the acan do to a function. If |a|>1, then it vertically stretches the function (looks taller & skinnier) If 0<|a|<1, then it vertically shrinks the function (looks shorter and fatter)
- 5. -areflects the function over the x axis.
- 6. Next, let’s look at what the hcan do to a function. If you have (x-h), the function is shifted right h units If you have (x+h), the function is shifted left h units
- 7. Next, let’s look at what the kcan do to a function. If you have (x)-k, the function is shifted down k units If you have (x)+k, the function is shifted up k units
- 8. A less common transformation is reflecting the function over the y axis (mirror image). To create a y axis reflection, you negate the x inside of it’s grouping symbols. For example: 𝑦 = (−𝑥)2 OR 𝑦 = −𝑥 OR 𝑦 = 2−𝑥
- 9. Two more less common transformations are horizontal stretching and horizontal shrinking. To create a horizontal change, you place a number in front of x inside of it’s grouping symbols. For example: 𝑦 = (2𝑥)3 or 𝑦 = 1 2 𝑥 or 𝑦 = 11𝑥
- 10. Specifics on horizontal stretching and horizontal shrinking: • If the coefficient is >1, then it horizontally shrinks the function • If the coefficient is between 0 and 1, then it horizontally stretches the function.
- 11. All of those transformation rules work on any of our functions! Let’s try an example. Given 𝑓 𝑥 = 𝑥3, shift it up 10 and left 7, reflect it across the x axis, and vertically stretch it by 5. • 𝑓 𝑥 = 𝑥3 + 10 now it’s shifted up 10 • 𝑓 𝑥 = (𝑥 + 7)3+10 now we’ve added the shift left 7 • 𝑓 𝑥 = −(𝑥 + 7)3 + 10 now we’ve reflected it over the x axis • 𝑓 𝑥 = −5(𝑥 + 7)3 + 10 lastly, we vertically stretched it by 5
- 12. What if it already has transformations and we’re asked to change it? Let’s try that. Given 𝑓 𝑥 = 7(2 𝑥) − 3, shift it up 4 and reflect it across the y axis. • 𝑓 𝑥 = 7 2 𝑥 − 3 + 4 this shifts it up 4, but we should combine like terms • 𝑓 𝑥 = 7 2 𝑥 + 1 now we just need to negate the x • 𝑓 𝑥 = 7 2−𝑥 + 1 It’s done.
- 13. Here are some extra resources to help you: Group of video tutorials: • https://www.tes.com/lessons/x0Ei03pVKLMCxg/m2l6- transformations-of-functions Website with graph examples: • http://www.regentsprep.org/regents/math/algtrig/atp9/funclesso n1.htm • https://www.mathsisfun.com/sets/function-transformations.html