The quadratic function f is defined by an equation in the form \(\mathrm{f(x) = a(x + h)^2 + k}\), where...

1. TRANSLATE the given information

Given:

• \(\mathrm{f(x)}\) is a quadratic function in the form \(\mathrm{f(x) = a(x + h)^2 + k}\)
• From the graph, we can identify the vertex and other key features
• We need to find \(\mathrm{g(x) = f(x - 2)}\)

What we need:

• The equation for \(\mathrm{f(x)}\) first
• Then apply the transformation to get \(\mathrm{g(x)}\)

2. INFER the function f(x) from the graph

From the graph, identify:

• Vertex: \(\mathrm{(-3, -8)}\) → This tells us \(\mathrm{h = -3}\) and \(\mathrm{k = -8}\)
• The function has form: \(\mathrm{f(x) = a(x + 3)^2 - 8}\)

To find 'a', we'd use another point from the graph. Based on the answer choices, we determine:

• \(\mathrm{f(x) = 2(x + 3)^2 - 8}\)

3. TRANSLATE the transformation notation

The notation \(\mathrm{g(x) = f(x - 2)}\) means:

• Take the function \(\mathrm{f(x)}\)
• Replace every x with \(\mathrm{(x - 2)}\)

This is a horizontal shift of 2 units to the right.

4. SIMPLIFY to find g(x)

Start with: \(\mathrm{g(x) = f(x - 2)}\)

Substitute \(\mathrm{(x - 2)}\) into \(\mathrm{f(x)}\):

\(\mathrm{g(x) = 2((x - 2) + 3)^2 - 8}\)

Simplify inside the parentheses:

\(\mathrm{g(x) = 2(x - 2 + 3)^2 - 8}\)

\(\mathrm{g(x) = 2(x + 1)^2 - 8}\)

Answer: A. \(\mathrm{g(x) = 2(x + 1)^2 - 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting \(\mathrm{g(x) = f(x - 2)}\) as meaning "subtract 2 from the function" rather than "replace x with \(\mathrm{(x - 2)}\)."

Students might think: "If \(\mathrm{g(x) = f(x - 2)}\), I should just subtract 2 from \(\mathrm{f(x)}\)," leading them to write:

\(\mathrm{g(x) = 2(x + 3)^2 - 8 - 2 = 2(x + 3)^2 - 10}\)

This may lead them to select Choice D: \(\mathrm{g(x) = 2(x + 3)^2 - 10}\)

Second Most Common Error:

Weak INFER skill: Confusing the direction of horizontal shifts or incorrectly applying the transformation to the vertex.

Students might think: "\(\mathrm{g(x) = f(x - 2)}\) means shift left by 2, so the vertex moves from \(\mathrm{(-3, -8)}\) to \(\mathrm{(-5, -8)}\)," leading to:

\(\mathrm{g(x) = 2(x + 5)^2 - 8}\)

This may lead them to select Choice B: \(\mathrm{g(x) = 2(x + 5)^2 - 8}\)

Alternatively, students might incorrectly think the shift affects only the constant term or apply a vertical shift instead of horizontal, leading them toward Choice C: \(\mathrm{g(x) = 2(x + 3)^2 - 6}\)

The Bottom Line:

The critical challenge is correctly interpreting function transformation notation. The notation \(\mathrm{f(x - 2)}\) requires substitution throughout the entire function, not just modifying one part. Understanding that \(\mathrm{x \rightarrow (x - 2)}\) means every x gets replaced is the key conceptual leap. Additionally, remembering that the direction of horizontal shifts is counterintuitive (minus shifts right, plus shifts left) prevents common directional errors.