A quadratic function has a vertex at , where and are constants. In the -plane, the graph of contains the points and . The function is defined by , and its graph has a vertex at . If , what is the value of ?
Cevap: 8
Cevap
The value of k is 8.
The axis of symmetry of the quadratic function f is located at the midpoint of the x-values of any two points on the graph that have the same y-value. Since the graph of f contains (-1, 3) and (7, 3), the x-coordinate of its vertex, h, is equal to the midpoint (-1 + 7) / 2 = 3. Thus, the vertex of f is (3, k). The definition of g(x) = f(x + 2) - 4 indicates a horizontal translation of 2 units to the left and a vertical translation of 4 units down. Applying these translations to the vertex (3, k) of f gives the vertex of g as (3 - 2, k - 4) = (1, k - 4). Since the vertex of g is defined as (p, q), we have p = 1 and q = k - 4. Substituting these values into the equation p + q = 5 yields 1 + (k - 4) = 5, which simplifies to k - 3 = 5, or k = 8.
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Anahtar Kavram
Quadratic functions and graphs, including symmetry and translations.