Two Truths and a Lie: Remember, the function has a maximum and an axis of symmetry at x = 2. Two of the following functions could represent this scenario while one cannot. Determine which two could represent the scenario and which one cannot. Explain how you know and show all necessary work to support your reasoning. a) f(x) = −x 2 + 4x + 2 b) g(x) = −(x + 2) 2 − 5 c) h(x) = −2(x − 1)(x − 3)

Answer

Functions F(x) and H(x) could represent the scenario, while G(x) cannot.

Identify the functions that meet the requirements. F(x) and H(x) could represent the scenario, while G(x) cannot because it does not have a maximum.
Explain how you know this to be true. The equation for the axis of symmetry is x = -b/2a, where a is the coefficient of the x^2 term and b is the coefficient of the x term. The vertex of a parabola is located at the point (h,k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. If the parabola is facing downwards (a < 0), then the vertex is a maximum. For F(x), a = -1 and b = 4, so x = -4/-2 = 2, meaning the axis of symmetry is at x = 2. The vertex for F(x) is (2,6), which means that there is a maximum at (2,6). For H(x), we can use the equation h(x) = a(x-p)^2 + k, where p is the axis of symmetry and k is the y-coordinate of the vertex. Expanding this equation gives us h(x) = -2(x^2-4x+3) = -2(x-1)(x-3), so h(x) also has an axis of symmetry at x = 2. The vertex of H(x) is (2,2), meaning that there is a maximum at (2,2). G(x) is in the form y = a(x-h)^2 + k, where a = -1 and h = -2, but its vertex is (-2,-5), which is the minimum point rather than the maximum point.