Unit 3 Parent Functions And Transformations Homework 5 Answer Key: Exact Answer & Steps

Unit 3: Parent Functions and Transformations – Homework 5 Answer Key (and How to Use It)

Opening hook

You’re staring at the screen, the cursor blinking on the last question of Homework 5, and your brain is doing a weird loop: “I know the answer, but I can’t remember how to get there.On top of that, we’re all in it together. This isn’t just a list of answers; it’s a roadmap that shows why the numbers look the way they do and how to spot the same patterns in any problem. Grab a pen, and let’s turn those “I need help” moments into “aha!The good news? ” That feeling is familiar. ” moments.

What Is Unit 3: Parent Functions and Transformations?

When we talk about parent functions, we’re talking about the simplest, most basic shapes that form the backbone of algebraic graphs. Think of them as the “templates” that teachers ask students to transform: shift, stretch, reflect, compress. In practice, the most common parent functions are:

• Linear: (y = x)
• Quadratic: (y = x^2)
• Absolute value: (y = |x|)
• Exponential: (y = 2^x) (or (y = a^x) with (a>1))
• Logarithmic: (y = \log_a x) (or (y = \ln x))
• Sine, Cosine, Tangent, etc.

Homework 5 dives into the transformations of these parents—shifting them left/right/up/down, reflecting over axes, scaling vertically/horizontally. The key is to keep the core shape intact while adjusting its position or size Not complicated — just consistent. No workaround needed..

Why It Matters / Why People Care

Transformations are the language of graphing. If you can master them, you can:

1. Predict how a function will look without drawing every point.
2. Solve real‑world problems where data is shifted or scaled—finance, physics, biology.
3. Ace exams that test conceptual understanding rather than rote memorization.

When students skip the transformation rules, they end up memorizing a table of answers that only works for the textbook. That’s why this answer key is more than a cheat sheet; it’s a teaching tool that reinforces the logic behind each step It's one of those things that adds up..

How It Works (or How to Do It)

Below is a walk‑through of each question in Homework 5, paired with the logic that leads to the answer. We’ll break it down into three parts: identify the parent, apply the transformation, and verify the result.

### 1. Identify the Parent Function

Every transformed function can be rewritten in the form
(y = f(ax + h) + k)
or, for vertical transformations,
(y = a,f(x - h) + k).

• (a) = vertical stretch/compression (and sign for reflection).
• (h) = horizontal shift (right if +, left if –).
• (k) = vertical shift (up if +, down if –).

Tip: Look for the “core” part—something that looks like a basic shape, e.g., (x^2), (|x|), (\sin x).

### 2. Apply the Transformation

1. Horizontal shift: replace (x) with (x - h).
2. Vertical shift: add or subtract (k).
3. Vertical stretch/compression: multiply the whole function by (a).
4. Reflection: change the sign of (a) or replace (x) with (-x).

### 3. Verify

• Plug in a simple value (often 0 or 1).
• Check if the graph’s vertex, intercepts, or asymptotes match the transformed description.

Homework 5 – Question‑by‑Question Answer Key

Below are the exact answers, but more importantly, the reasoning that lands you there.

1. (y = 2(x - 3)^2 + 5)

• Parent: (y = x^2).
• Horizontal shift right 3: replace (x) with (x-3).
• Vertical stretch by 2: multiply by 2.
• Vertical shift up 5: add 5.
  Answer: Vertex at ((3, 5)). The parabola opens upward, twice as steep as the basic (x^2).

2. (y = -|x + 4| + 2)

• Parent: (y = |x|).
• Horizontal shift left 4: ((x+4)).
• Reflection over x‑axis: negative sign.
• Vertical shift up 2: +2.
  Answer: V‑shape with vertex at ((-4, 2)), pointing downward.

3. (y = 3\sin(x - \frac{\pi}{2}) + 1)

• Parent: (y = \sin x).
• Horizontal shift right (\frac{\pi}{2}).
• Vertical stretch by 3: amplitude becomes 3.
• Vertical shift up 1: midline at (y = 1).
  Answer: Period still (2\pi), amplitude 3, midline at 1.

4. (y = \log_2(x/5) - 4)

• Parent: (y = \log_2 x).
• Horizontal stretch by factor 5: replace (x) with (x/5).
• Vertical shift down 4: (-4).
  Answer: Passes through ((5, -4)), asymptote at (x = 0).

5. (y = -5^{x-2})

• Parent: (y = 5^x).
• Horizontal shift right 2: ((x-2)).
• Reflection over x‑axis: negative sign.
  Answer: Exponential decay starting at (x = 2), crossing (y = -1) when (x = 2).

6. (y = \frac{1}{2}x + 3)

• Parent: (y = x).
• Vertical stretch/compression by (\frac{1}{2}): slope becomes (0.5).
• Vertical shift up 3: (+3).
  Answer: Line passes through ((0,3)) with slope (0.5).

7. (y = \frac{1}{x-1} + 2)

• Parent: (y = \frac{1}{x}).
• Horizontal shift right 1: replace (x) with (x-1).
• Vertical shift up 2: +2.
  Answer: Hyperbola with vertical asymptote (x = 1), horizontal asymptote (y = 2).

8. (y = \tan(x + \frac{\pi}{4}) - 1)

• Parent: (y = \tan x).
• Horizontal shift left (\frac{\pi}{4}).
• Vertical shift down 1: (-1).
  Answer: Period still (\pi), asymptotes shifted left (\frac{\pi}{4}).

9. (y = \sqrt{x - 6} - 3)

• Parent: (y = \sqrt{x}).
• Horizontal shift right 6: ((x-6)).
• Vertical shift down 3: (-3).
  Answer: Domain (x \ge 6), vertex at ((6, -3)).

10. (y = -\frac{1}{x^2 + 1})

• Parent: (y = \frac{1}{x^2}).
• Vertical reflection: negative sign.
• No shifts.
  Answer: Even function, maximum at (x = 0) where (y = -1).

Common Mistakes / What Most People Get Wrong

1. Mixing up horizontal and vertical shifts – forgetting that (x - h) shifts right, not up.
2. Forgetting the sign of (a) – a negative (a) reflects over the x‑axis, not the y‑axis.
3. Treating (\frac{1}{x-1}) as a simple shift – it’s a horizontal shift and a reciprocal, so the asymptote moves with it.
4. Misapplying the order of operations – always apply shifts inside the function first, then scale, then shift vertically.
5. Ignoring domain restrictions – especially with square roots and reciprocals; the transformed domain matters.

Practical Tips / What Actually Works

• Draw a quick sketch of the parent before transforming. It’s a visual cue that keeps you from losing track.
• Write the transformation in steps:
  1. Start with (y = f(x)).
  2. Apply horizontal shift: (f(x - h)).
  3. Apply vertical stretch/compress: (a f(x - h)).
  4. Apply vertical shift: (a f(x - h) + k).
• Check special points (vertex, intercepts, asymptotes) after each transformation.
• Use color coding if you’re drawing on paper: one color for the parent, another for each shift.
• Practice with “inverse” problems: given a transformed graph, write the equation. This cements the logic.

FAQ

Q1: How do I remember the sign of horizontal shifts?
A1: Think of the parent graph moving in the opposite direction of the sign inside. (x - h) pushes right; (x + h) pushes left.

Q2: What’s the difference between a vertical stretch and a horizontal stretch?
A2: A vertical stretch multiplies the output (y) (the height). A horizontal stretch divides the input (x) (the width).

Q3: Why does (\log_a(x/b)) shift horizontally instead of vertically?
A3: Because the argument of the log is scaled. Scaling inside the log changes the x‑axis, not the y‑axis.

Q4: Can I combine transformations in any order?
A4: The order matters for shifts but not for scaling if you’re only doing vertical stretches. Always start with shifts inside the function, then scale, then shift vertically.

Q5: How do I check if my answer is correct?
A5: Plug in a few easy x‑values (0, 1, -1) and see if the y‑values line up with the described shifts and stretches.

Closing paragraph

So there you have it: a clear, step‑by‑step key that not only tells you the correct answers for Homework 5 but also gives you the intuition to tackle any parent‑function transformation that comes your way. Keep this framework in mind, practice a handful of new problems, and soon those “I’m stuck” moments will turn into confident, quick solves. Happy graphing!