When the quadratic function f𝑓 is graphed in the xy𝑥𝑦-plane, where y=f(x)𝑦=𝑓(𝑥), its vertex is (−4, −2)(-4, -2). One of the x𝑥-intercepts of this graph is (−274,0)(-274,0). What is the other x𝑥-intercept of the graph?(−192,0)(-192,0)eliminate(−114,0)(-114,0)eliminate(−94,0)(-94,0)eliminate(−54,0)
Question
When the quadratic function f𝑓 is graphed in the xy𝑥𝑦-plane, where y=f(x)𝑦=𝑓(𝑥), its vertex is (−4, −2)(-4, -2). One of the x𝑥-intercepts of this graph is (−274,0)(-274,0). What is the other x𝑥-intercept of the graph?(−192,0)(-192,0)eliminate(−114,0)(-114,0)eliminate(−94,0)(-94,0)eliminate(−54,0)
Solution
The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Given that the vertex is (-4, -2), we can write the function as f(x) = a(x+4)^2 - 2.
We also know that one of the x-intercepts is (-274, 0). Substituting these values into the equation gives us 0 = a(-274+4)^2 - 2. Solving for a gives us a = 2/(-270)^2.
Now, the x-intercepts of a parabola are the values of x for which f(x) = 0. Setting f(x) = 0 and solving for x gives us x = -4 ± sqrt(2/a). Substituting a = 2/(-270)^2 into this equation gives us x = -4 ± sqrt(-270^2/2).
We already know that one of the x-intercepts is -274, so the other x-intercept must be -4 + sqrt(-270^2/2) = -4 + 270 = 266. Therefore, the other x-intercept of the graph is (266, 0).
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