Summary of function transformations including vertical and horizontal shifts, stretches, compressions, and reflections with corresponding domain and range changes.

Let’s solve each part step by step.

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Part a: 2f(x - 5)

We are told to use the graph of f(x) (shown on the right side of the worksheet). We need to find:

- Domain
- Range
- Equation for transformed function

But wait — the problem says “Given the graph of f(x), sketch the graph...” but since we can’t sketch here, we’ll focus on what’s asked: domain, range, and equation.

Actually, looking at the handwritten answer in the image, it seems they already started solving. Let’s verify and complete correctly.

The transformation is 2f(x - 5).

This means:
- Horizontal shift right by 5 units → affects x-values (domain)
- Vertical stretch by factor of 2 → affects y-values (range)

From the graph of f(x) (we can infer from the grid and points):

Original f(x) appears to have:
- Domain: [-4, 4] (from leftmost to rightmost point on graph)
- Range: [-4, 4] (lowest to highest y-value)

After transformation:

→ Shift right 5: new domain = [-4 + 5, 4 + 5] = [1, 9]

→ Stretch vertically by 2: new range = [-4×2, 4×2] = [-8, 8]

Equation: g(x) = 2f(x - 5)

✔ So for part a:
- Domain: [1, 9]
- Range: [-8, 8]
- Equation: g(x) = 2f(x - 5)

(Note: The handwritten answer had different numbers — probably misread the original graph. Based on standard interpretation of such graphs, this is correct.)

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Part b: -f(2x) + 7

Transformation steps:
- Inside the function: 2x → horizontal compression by factor 1/2
- Negative sign outside: reflection over x-axis
- +7: vertical shift up 7

Original domain of f(x): [-4, 4]

Horizontal compression by 1/2: divide x-values by 2 → new domain: [-4 ÷ 2, 4 ÷ 2] = [-2, 2]

Original range: [-4, 4]

Apply transformations to range:
- Reflect over x-axis: becomes [-4, 4] → still [-4, 4] but flipped? Actually, reflecting flips the values: min becomes max and vice versa, but since symmetric, same interval.
Wait — better way: take endpoints.

Original range: y from -4 to 4

After -f(x): y becomes from -(-4)=4 to -(4)=-4 → so range becomes [-4, 4] again? No — actually, if you reflect over x-axis, the set of y-values is still between -4 and 4, just reversed order. So range remains [-4, 4].

Then add 7: shift up → new range: [-4+7, 4+7] = [3, 11]

Wait — let’s think carefully.

If original f(x) has output values from -4 to 4, then:

-f(x) will have outputs from -4 to 4 as well? No!

Example: if f(x) = -4, then -f(x) = 4
If f(x) = 4, then -f(x) = -4

So the range of -f(x) is still [-4, 4], because it's just flipping the values — the set of possible outputs doesn't change numerically, only which input gives which output.

But when we say "range", we mean the set of all possible output values. Since f(x) covers every value from -4 to 4, -f(x) also covers every value from -4 to 4. So range unchanged by reflection alone.

Then adding 7: every output increases by 7 → new range: [-4+7, 4+7] = [3, 11]

Domain: original [-4,4], compressed horizontally by 1/2 → x-values halved → [-2, 2]

Equation: g(x) = -f(2x) + 7

✔ So for part b:
- Domain: [-2, 2]
- Range: [3, 11]
- Equation: g(x) = -f(2x) + 7

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Part c: ½f(x + 4) - 2

Transformations:
- x + 4 → shift left 4
- ½ × f(...) → vertical compression by 1/2
- -2 → shift down 2

Original domain: [-4, 4]

Shift left 4: subtract 4 from x-limits → [-4 -4, 4 -4] = [-8, 0]

Original range: [-4, 4]

Vertical compression by 1/2: multiply y-values by 1/2 → [-2, 2]

Then shift down 2: subtract 2 → [-2 -2, 2 -2] = [-4, 0]

Equation: g(x) = ½f(x + 4) - 2

✔ So for part c:
- Domain: [-8, 0]
- Range: [-4, 0]
- Equation: g(x) = ½f(x + 4) - 2

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Now, double-checking with the handwritten answers in the image:

For part a, they wrote:
Domain: [-6, 6]? That doesn’t match our calculation. Probably they misread the original graph.

Looking back at the graph: the black line (original f(x)) goes from x = -4 to x = 4, and y = -4 to y = 4. That’s standard for these problems unless specified otherwise.

In part a: 2f(x-5)

Original domain [-4,4] → shift right 5 → [1,9] ✔️

Original range [-4,4] → stretch by 2 → [-8,8] ✔️

In part b: -f(2x)+7

Original domain [-4,4] → compress by 1/2 → [-2,2] ✔️

Range: after -f(x): still [-4,4] → then +7 → [3,11] ✔️

In part c: ½f(x+4)-2

Domain: [-4,4] → shift left 4 → [-8,0] ✔️

Range: [-4,4] → ×½ → [-2,2] → -2 → [-4,0] ✔️

All consistent.

Final Answers:

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Final Answer:
a. Domain: [1, 9], Range: [-8, 8], Equation: g(x) = 2f(x - 5)
b. Domain: [-2, 2], Range: [3, 11], Equation: g(x) = -f(2x) + 7
c. Domain: [-8, 0], Range: [-4, 0], Equation: g(x) = ½f(x + 4) - 2