Let's correct the vertical shift based on the instruction to shift the graph vertically by \(-b\) units.

### Part (a): Transformations (Corrected)

You need to find the new function after the following transformations:
1. **Shift horizontally to the left by \(a\) units**.
2. **Shift vertically by \(-b\) units**.
3. **Reflect across both axes**.

#### Step 1: Horizontal Shift
Shifting the graph to the left by \(a\) units means replacing \(x\) with \((x + a)\). The function becomes:
\[
y = (x + a)^2 - 4(x + a) + 5
\]

Expanding and simplifying:
\[
y = (x^2 + 2ax + a^2) - 4x - 4a + 5
\]
\[
y = x^2 + 2ax - 4x + a^2 - 4a + 5
\]
\[
y = x^2 + (2a - 4)x + (a^2 - 4a + 5)
\]

#### Step 2: Vertical Shift (by \(-b\) units)
Shifting vertically by \(-b\) units means subtracting \(b\) from the function:
\[
y = x^2 + (2a - 4)x + (a^2 - 4a + 5 - b)
\]

#### Step 3: Reflection
Reflecting across both axes involves:
- **Reflection across the y-axis**: Replace \(x\) with \(-x\).
- **Reflection across the x-axis**: Multiply the entire function by \(-1\).

Applying both reflections:
\[
y = -[(-x)^2 + (2a - 4)(-x) + (a^2 - 4a + 5 - b)]
\]
Simplifying:
\[
y = -[x^2 - (2a - 4)x + (a^2 - 4a + 5 - b)]
\]
\[
y = -x^2 + (2a - 4)x - (a^2 - 4a + 5 - b)
\]

So, the transformed function is:
\[
y = -x^2 + (2a - 4)x - (a^2 - 4a + 5 - b)
\]

### Part (b): Sketch the Graph
To sketch the graph, you would substitute specific values for \(a\) and \(b\). If \(a = 3\) and \(b = 2\), the function becomes:
\[
y = -x^2 + (2(3) - 4)x - (3^2 - 4(3) + 5 - 2)
\]
Simplify this:
\[
y = -x^2 + 2x - (9 - 12 + 3) = -x^2 + 2x
\]
So, the equation for this specific graph would be:
\[
y = -x^2 + 2x - 0
\]
You would then plot this quadratic function, noting that it opens downward because the leading coefficient is negative.

### Part (c): Solve for \(A + B - C\)
Given the transformed function is of the form \(y = Ax^2 + Bx + C\), we have:
\[
A = -1, \quad B = 2, \quad C = 0
\]
Now, compute \(A + B - C\):
\[
A + B - C = -1 + 2 - 0 = 1
\]

### Summary
- **(a)** The transformed function after the shifts and reflections is \(y = -x^2 + (2a - 4)x - (a^2 - 4a + 5 - b)\).
- **(b)** The specific graph for \(a = 3\) and \(b = 2\) is \(y = -x^2 + 2x\).
- **(c)** The value of \(A + B - C\) after the transformation is 1.

### Questions
1. What effect would shifting the graph vertically by \(+b\) units have on the function?
2. How does a horizontal shift to the right differ from a shift to the left?
3. If \(a = 0\) and \(b = 0\), what is the resulting transformed function?
4. What are the vertex and axis of symmetry of the transformed function?
5. How does the sign of the coefficient \(A\) affect the direction of the parabola?

### Tip
Always pay careful attention to the direction of the shift, whether it’s positive or negative, as it directly affects the function’s translation.

Learn how to transform quadratic functions by shifting them vertically by -b units. This detailed solution includes steps for horizontal and vertical shifts, reflection across axes, and practical examples. Suitable for advanced high school students.

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