Practice worksheet on transforming linear functions with graph analysis and transformation rules.

Let’s go step by step through the problems on this worksheet. Since you didn’t specify which problem to solve, I’ll assume you want help with all of them — but since that’s too long for one response, I’ll pick Problem 8, which is a real-world application and often tricky. If you meant a different problem, just say so!

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Problem 8: The Red Cab Taxi Service charges $1.00 for the first 1/4 mile and $0.75 for each additional 1/4 mile. The company just raised its rates by a factor of 1.5.

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

First, let’s understand the original cost function f(x), where x = number of miles traveled.

- For the first 1/4 mile (or any distance ≤ 0.25 miles), it costs $1.00.
- After that, every additional 1/4 mile costs $0.75.

So if you travel x miles:

- Number of quarter-miles in x miles = x / 0.25 = 4x
- First quarter-mile: $1.00
- Remaining quarter-miles: (4x - 1) → but only if x > 0.25

Actually, better to think in terms of how many *additional* quarter-miles beyond the first.

Let’s define:

If x ≤ 0.25 → cost = $1.00
If x > 0.25 → cost = 1.00 + 0.75 × (number of additional quarter-miles)

Number of additional quarter-miles = total quarter-miles minus 1 = (x / 0.25) - 1 = 4x - 1

But wait — actually, since we’re dealing with continuous distance, not discrete chunks, we need to use ceiling or floor? Actually, no — in math problems like this, unless specified otherwise, we usually treat it as linear after the first segment.

Wait — looking at typical textbook treatments, they often model this as:

f(x) = 1.00 + 0.75 * (4*(x - 0.25)) for x ≥ 0.25

Simplify:
= 1.00 + 3*(x - 0.25)
= 1.00 + 3x - 0.75
= 3x + 0.25 for x ≥ 0.25

And for x < 0.25, f(x) = 1.00

But actually, even simpler: since 1/4 mile increments are used, maybe we should express it piecewise.

However, many textbooks simplify this to a single expression assuming x ≥ 0.25, or sometimes they write:

f(x) = 1.00 + 0.75 * ceil(4x - 1) — but that’s messy.

Alternatively, perhaps the problem expects us to ignore the “first 1/4 mile” nuance and just say:

Total cost = base fare + rate per extra 1/4 mile.

But here’s a better way: Let’s define the function based on distance in miles.

Let’s suppose the taxi charges:

- $1.00 for up to 0.25 miles
- Then $0.75 for every 0.25 miles after that.

So for x miles:

Cost = 1.00 + 0.75 * max(0, (x - 0.25)/0.25 )

Because (x - 0.25)/0.25 gives the number of additional quarter-miles.

Simplify: (x - 0.25)/0.25 = 4(x - 0.25) = 4x - 1

So:

f(x) = 1.00 + 0.75 * max(0, 4x - 1)

Now, the company raises rates by a factor of 1.5 → multiply entire function by 1.5.

So g(x) = 1.5 * f(x) = 1.5 * [1.00 + 0.75 * max(0, 4x - 1)]

We can distribute:

g(x) = 1.5*1.00 + 1.5*0.75 * max(0, 4x - 1)
g(x) = 1.50 + 1.125 * max(0, 4x - 1)

That’s exact, but maybe we leave it like that.

Alternatively, if we want to write it without max, piecewise:

For x ≤ 0.25:
g(x) = 1.5 * 1.00 = 1.50

For x > 0.25:
g(x) = 1.5 * [1.00 + 0.75*(4x - 1)]
= 1.5 * [1.00 + 3x - 0.75]
= 1.5 * [3x + 0.25]
= 4.5x + 0.375

So final answer for part a:

g(x) =
{
1.50, if x ≤ 0.25
4.5x + 0.375, if x > 0.25
}

But maybe the problem expects a simplified version ignoring the piecewise nature? Or perhaps they consider the "first 1/4 mile" included in the scaling.

Another approach: Maybe they mean the entire pricing structure is scaled — so instead of $1.00 start and $0.75 per 1/4 mile, now it's $1.50 start and $1.125 per 1/4 mile.

Which matches what we have.

So yes, g(x) = 1.5 * f(x)

Part b: Describe the transformation(s).

Since g(x) = 1.5 * f(x), this is a vertical stretch by a factor of 1.5.

No horizontal shift, no reflection — just stretching the output values by 1.5 times.

So transformations: Vertical stretch by factor of 1.5.

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✔ Final Answer for Problem 8:

a. g(x) = 1.5 * f(x), where f(x) is the original cost function. Specifically:
- If x ≤ 0.25, g(x) = 1.50
- If x > 0.25, g(x) = 4.5x + 0.375

b. The transformation is a vertical stretch by a factor of 1.5.

But since the question says “write a new price function”, and doesn’t specify format, we can also write:

g(x) = 1.5 \cdot \left(1.00 + 0.75 \cdot \max\left(0, \frac{x - 0.25}{0.25}\right)\right)

Or simplified:

g(x) = 1.5 + 1.125 \cdot \max(0, 4x - 1)

But perhaps the cleanest is to state:

g(x) = 1.5 \times (\text{original cost})

And for description: vertical stretch by factor of 1.5.

Given that, I’ll box the key answers.

Final Answer:
a. \( g(x) = 1.5 \cdot f(x) \), where \( f(x) \) is the original fare function. Explicitly:
For \( x \leq 0.25 \), \( g(x) = 1.50 \);
For \( x > 0.25 \), \( g(x) = 4.5x + 0.375 \).
b. Vertical stretch by a factor of 1.5.