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Describe the transformation of f(x) = x² represented by g (x) = (x - 1)² - 2. Then identify the graph of each function

The graph of g is a translation _____________ and _____________ of the graph of f.

Answer choices
⬇️
1 Unit left, 1 Unit right, 1 Unit up, 1 Unit down,
2 units left, 2 units right, 2 units up, 2 units down

1 answer

To analyze the transformation of the function \( f(x) = x^2 \) to \( g(x) = (x - 1)^2 - 2 \), we can break it down step by step.

  1. Starting function: \( f(x) = x^2 \)

    • This is a parabola that opens upwards with the vertex at (0, 0).

  2. Step 1 - Horizontal Translation: \( (x - 1)^2 \)

    • The expression \( (x - 1) \) indicates a horizontal translation of the graph to the right by 1 unit.
    • So, after this step, the vertex of the parabola moves from (0, 0) to (1, 0).

  3. Step 2 - Vertical Translation: \(-2\)

    • The subtraction of 2 in \(-2\) indicates a vertical translation downwards by 2 units.
    • Thus, the vertex moves from (1, 0) to (1, -2).

Putting these transformations together, the graph of \( g(x) = (x - 1)^2 - 2 \) is obtained by first shifting the graph of \( f(x) = x^2 \) to the right by 1 unit and then down by 2 units.

So, the final answer to the fill-in-the-blank statement is:

The graph of \( g \) is a translation 1 unit right and 2 units down of the graph of \( f \).