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The graph of h as a translation of the graph of f(x) = x²

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The graph of h as a translation of the graph of f(x) = x²

The graph of h is the graph of g(x) = (x – 2)² + 6 translated 5 units left and 3 units down.
a. Describe the graph of h as a translation of the graph of f(x) = x².
b. Write the function h in vertex form.

a. The graph of h is the graph of f(x) = x² translated ___ unit(s) ___ and ___ unit(s) ___.
(Type whole numbers.)




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    1 Answer

    1. Quizzma Team

      To solve the problem, let’s break it down step by step.

      ### a. Describe the graph of h as a translation of the graph of f(x) = x².

      1. Starting Function: The graph of f(x) = x² has its vertex at the origin (0, 0).
      2. Translation: The function g(x) = (x – 2)² + 6 has its vertex at (2, 6). To get from f(x) to g(x), we shift the graph 2 units to the right and 6 units up.
      3. Final Translation to h: The graph of h is then translated 5 units left and 3 units down from g(x):

      – Moving 5 units left from (2, 6) brings us to (-3, 6).

      – Moving 3 units down from (−3, 6) gives us the new vertex at (−3, 3).

      Thus, from f(x) to h:

      – Translation: The graph of h is the graph of f(x) = x² translated 3 units down and 5 units left.

      ### b. Write the function h in vertex form.

      1. Vertex from Translation: The new vertex for h is (-3, 3).
      2. Vertex Form: The vertex form of a quadratic function is h(x) = a(x – h)² + k, where (h, k

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