#AP Pre-Calculus: Transformations of Functions - Your Ultimate Review 🚀

Hey there! Let's make sure you're totally prepped for transformations. This guide is designed to be your best friend tonight, making everything crystal clear and helping you feel confident for the exam tomorrow. Let’s dive in!

#1.12 Transformations of Functions

Transformations are all about how we can move, stretch, or flip functions. Think of it like playing with Play-Doh – you can shift it, squish it, or mirror it! We’ll cover additive (translations) and multiplicative (dilations and reflections) transformations. Let's get started!

# Additive Transformations (Translations) 🚶‍♀️🚶‍♂️

Additive transformations involve adding or subtracting constants, which causes the function to shift its position on the graph. These are also known as translations.

#1️⃣ Vertical Translations ⬆️⬇️

• Concept: $g(x) = f(x) + k$ shifts the graph of $f(x)$ vertically by $k$ units.
  • If $k > 0$, the graph moves up.
  • If $k < 0$, the graph moves down.
• Key Point: The shape of the graph stays the same; only its position changes.

Image Courtesy of Cuemath

Caption: A graph of the function f(x) and its new graph g(x) that’s shifted 3 units upward

#2️⃣ Horizontal Translations ↔️

• Concept: $g(x) = f(x + h)$ shifts the graph of $f(x)$ horizontally by $h$ units.
  • If $h > 0$, the graph moves to the left.
  • If $h < 0$, the graph moves to the right.
• Key Point: Remember, it's the opposite of what you might expect with the sign inside the function.

Image Courtesy of Quora

Caption: A graph of the function f(x) and its new graph g(x) that’s shifted six units to the left

# 🪞 Multiplicative Transformations (Dilations and Reflections) 🪞

Multiplicative transformations involve multiplying the function by a constant, which causes the function to stretch, shrink, or flip. These include dilations and reflections.

#1️⃣ Vertical Dilations ↕️

• Concept: $g(x) = af(x)$ scales the graph of $f(x)$ vertically by a factor of $|a|$.
  • If $|a| > 1$, the graph is stretched vertically (taller).
  • If $0 < |a| < 1$, the graph is shrunk vertically (shorter).
  • If $a < 0$, the graph is also reflected over the x-axis. 🔁

Image Courtesy of Github

Caption: A function graph of $4x^2$ and its new function graph $1/4x^2$ that appears shorter

#2️⃣ Horizontal Dilations ↔️

• Concept: $g(x) = f(bx)$ scales the graph of $f(x)$ horizontally by a factor of $|1/b|$.
  • If $|b| > 1$, the graph is shrunk horizontally (wider).
  • If $0 < |b| < 1$, the graph is stretched horizontally (narrower).
  • If $b < 0$, the graph is also reflected over the y-axis. 🔁

Image Courtesy of Nagwa

Caption: A function graph and its new function graph that’s stretched horizontally appearing narrower.

#Combining Transformations 🤯

• General Form: $g(x) = a \cdot f(bx + h) + k$
  • a : Vertical dilation and reflection
  • b : Horizontal dilation and reflection
  • h : Horizontal translation
  • k : Vertical translation
• Order Matters: Apply horizontal transformations before vertical ones.

#Domain and Range Changes 🧐

• Transformations can affect the domain and range of a function.
  • Reflections can change the direction of the range.
  • Dilations can compress or expand the domain and range.

Image Courtesy of Lumen Learning

Caption: A coordinate plane with eight functions plotted

# Final Exam Focus 🎯

Okay, let's get down to what really matters for the exam. Here's what you absolutely need to nail:

• High-Priority Topics:
  • Combining multiple transformations in a single problem. 🤯
  • Identifying transformations from equations and graphs. 🧐
  • Understanding how transformations affect domain and range. 😮
• Common Question Types:
  • Multiple-choice questions asking you to identify transformations from a given equation or graph.
  • Free-response questions that require you to apply a series of transformations to a function.
  • Questions that ask you to write the equation of a transformed function given its graph.

• Time Management:
  • Quickly sketch the parent function and then apply the transformations step-by-step.
  • Use process of elimination in multiple-choice questions to save time.
• Common Pitfalls:
  • Mixing up horizontal and vertical transformations.
  • Misinterpreting the signs of h and b.
  • Forgetting to consider the impact of reflections.

# Practice Questions 📝

Here are some practice questions to test your understanding. Remember, practice makes perfect!

Practice Question

#Multiple Choice Questions

1. The graph of $y = f(x)$ is transformed to $y = -2f(x-3) + 1$. Which of the following describes the transformations? (A) Vertical stretch by 2, reflection over the x-axis, shift right 3, shift up 1 (B) Vertical compression by 1/2, reflection over the x-axis, shift left 3, shift down 1 (C) Vertical stretch by 2, reflection over the y-axis, shift right 3, shift up 1 (D) Vertical stretch by 2, reflection over the x-axis, shift left 3, shift up 1

2. If the function $f(x)$ has a domain of $[-5, 5]$ and a range of $[-2, 8]$, what is the domain and range of $g(x) = f(2x)$? (A) Domain: $[-10, 10]$, Range: $[-2, 8]$ (B) Domain: $[-5, 5]$, Range: $[-4, 16]$ (C) Domain: $[-2.5, 2.5]$, Range: $[-2, 8]$ (D) Domain: $[-5, 5]$, Range: $[-1, 4]$

3. The graph of $y=f(x)$ is shown below. Which of the following represents the graph of $y=f(-x)$?

   (A) The graph of $f(x)$ reflected across the y-axis. (B) The graph of $f(x)$ reflected across the x-axis. (C) The graph of $f(x)$ shifted 1 unit to the right. (D) The graph of $f(x)$ shifted 1 unit to the left.

#Free Response Question

Consider the function $f(x) = |x|$.

(a) Sketch the graph of $f(x)$. (1 point)

(b) Sketch the graph of $g(x) = -2f(x+1) - 3$. (3 points)

(c) State the domain and range of $g(x)$. (2 points)

Scoring Rubric:

(a) - 1 point: Correct graph of $f(x) = |x|$.

(b) - 1 point: Correct horizontal shift to the left by 1 unit. - 1 point: Correct vertical stretch by a factor of 2 and reflection over the x-axis. - 1 point: Correct vertical shift down by 3 units.

(c) - 1 point: Correct domain (all real numbers). - 1 point: Correct range $(-\infty, -3]$.

That's it! You've got this. Go into the exam with confidence, and remember all the cool secrets we've unlocked tonight. You're going to do great! 🎉