Transforming Linear Functions Worksheet: Complete with ease ... - Free Printable

You're working on “Practice B: Transforming Linear Functions” from a Holt McDougal Algebra 2 worksheet. Since I can’t see the image, I’ll solve all problems based on standard textbook content and the visible text you provided.

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## 🔢 Problem 1:
> *Let g(x) be the indicated transformation of f(x). Write the rule for g(x).*

Given:

- f(x) = x
- Transformation: horizontal translation left 3 units

✔ Solution:
To translate a function left by 3 units, replace x with (x + 3).

So,
> g(x) = f(x + 3) = x + 3

✔️ Answer: `g(x) = x + 3`

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## 🔢 Problem 2:
> *vertical compression by a factor of ½*

✔ Solution:
A vertical compression by a factor of ½ means multiplying the output (y-values) by ½.

So,
> g(x) = ½ · f(x) = ½x

✔️ Answer: `g(x) = ½x`

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## 🔢 Problem 3:
> *reflection across the y-axis*

✔ Solution:
Reflection across the y-axis replaces x with –x.

So,
> g(x) = f(–x) = –x

✔️ Answer: `g(x) = –x`

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## 🔢 Problem 4:
> *linear function defined by the table; horizontal stretch by a factor of 2.3*

Given table:

| x | –5 | 0 | 7 |
|----|----|----|----|
| f(x) | –3 | 2 | 21 |

We need to find the rule for f(x) first.

✔ Step 1: Find slope (m)

Use two points: (0, 2) and (7, 21)

> m = (21 – 2)/(7 – 0) = 19/7

✔ Step 2: Use point-slope or slope-intercept

Since it passes through (0, 2), y-intercept is 2.

> f(x) = (19/7)x + 2

✔ Step 3: Apply horizontal stretch by factor of 2.3

Horizontal stretch by factor k means replace x with x/k

So,

> g(x) = f(x / 2.3) = (19/7)(x / 2.3) + 2

Simplify:

> g(x) = (19x)/(7 × 2.3) + 2

Compute denominator: 7 × 2.3 = 16.1

> g(x) = (19/16.1)x + 2

You can leave as fraction or decimal:

As decimal: 19 ÷ 16.1 ≈ 1.1801

But better to write as exact fraction:

Note: 2.3 = 23/10 → so 7 × 23/10 = 161/10

Then:

> g(x) = (19/7) × (x / (23/10)) + 2 = (19/7) × (10x/23) + 2 = (190x)/(161) + 2

✔️ Answer: `g(x) = (190/161)x + 2`

*(Alternatively, if decimal is acceptable: `g(x) ≈ 1.180x + 2`)*

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## 🔢 Problem 5:
> *f(x) = 1.7x – 3; vertical compression by a factor of 0.7*

✔ Solution:
Vertical compression by 0.7 → multiply entire function by 0.7

> g(x) = 0.7 × f(x) = 0.7(1.7x – 3) = (0.7 × 1.7)x – (0.7 × 3)

Calculate:

- 0.7 × 1.7 = 1.19
- 0.7 × 3 = 2.1

> g(x) = 1.19x – 2.1

✔️ Answer: `g(x) = 1.19x – 2.1`

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## 🔢 Problem 6:
> *vertical translation down 2 units followed by a horizontal compression by a factor of ⅔*

Start with f(x) = x

✔ Step 1: Vertical translation down 2 units

> f₁(x) = x – 2

✔ Step 2: Horizontal compression by factor of ⅔

Horizontal compression by factor k (where k < 1) → replace x with x / k

Here, k = ⅔ → so replace x with x / (⅔) = x × ³⁄₂

> g(x) = f₁( (3/2)x ) = (3/2)x – 2

✔️ Answer: `g(x) = (3/2)x – 2`

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## 🔢 Problem 7:
> *horizontal stretch by a factor of 3.2 followed by a horizontal translation right 1 unit*

Start with f(x) = x

✔ Step 1: Horizontal stretch by 3.2

Replace x with x / 3.2

> f₁(x) = x / 3.2

✔ Step 2: Horizontal translation right 1 unit

Replace x with x – 1

> g(x) = f₁(x – 1) = (x – 1) / 3.2

You can write as:

> g(x) = (1/3.2)(x – 1)

Or simplify 1/3.2 = 10/32 = 5/16

> g(x) = (5/16)(x – 1)

✔️ Answer: `g(x) = (5/16)(x – 1)`
*(or `g(x) = (x – 1)/3.2`)*

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## 🚖 Problem 8: Real-World Application — Red Cab Taxi Company

> The Red Cab Taxi Service charges $1.00 for the first ¼ mile and $0.75 for each additional ¼ mile. The company just raised its rates by a factor of 1.5.

Part a: Write a new price function g(x) for taxi rides.

✔ Step 1: Define original function f(x)

Let x = number of ¼ miles traveled.

- First ¼ mile: $1.00
- Each additional ¼ mile: $0.75

So for x ≥ 1:

> f(x) = 1.00 + 0.75(x – 1)

Simplify:

> f(x) = 1 + 0.75x – 0.75 = 0.75x + 0.25

Check: at x=1 → 0.75(1)+0.25 = 1.00 ✔
at x=2 → 0.75(2)+0.25 = 1.75 ✔ (first ¼ mi: $1, second ¼ mi: $0.75 → total $1.75)

So original function:
> f(x) = 0.75x + 0.25 for x ≥ 1 (and we can define f(x)=1 for 0<x<1, but since rate applies per ¼ mile, we assume x is integer ≥1)

✔ Step 2: Raise rates by factor of 1.5

This is a vertical stretch by 1.5, so:

> g(x) = 1.5 × f(x) = 1.5(0.75x + 0.25)

Compute:

> g(x) = 1.5×0.75x + 1.5×0.25 = 1.125x + 0.375

✔️ Answer (a): `g(x) = 1.125x + 0.375`

*(Where x = number of ¼-mile increments, x ≥ 1)*

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Part b: Describe the transformation(s) that have been applied.

✔ Answer:

The original price function f(x) was multiplied by 1.5, which is a vertical stretch by a factor of 1.5. This increases all prices by 50%.

No horizontal transformations were applied — only the vertical scaling.

✔️ Answer (b): A vertical stretch by a factor of 1.5 has been applied to the original price function.

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## ✔ Final Answers Summary:

1. `g(x) = x + 3`
2. `g(x) = ½x`
3. `g(x) = –x`
4. `g(x) = (190/161)x + 2`
5. `g(x) = 1.19x – 2.1`
6. `g(x) = (3/2)x – 2`
7. `g(x) = (5/16)(x – 1)`
8a. `g(x) = 1.125x + 0.375`
8b. Vertical stretch by a factor of 1.5

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