Which function has a greater y-intercept? f(x)=x^2-2x+12 Choose 1 answer: A f has a greater y-intercept than g. B g has a greater y -intercept than f. C f and g share the same y-intercept.

To determine which function has a greater \(y\)-intercept, we need to compare the \(y\)-intercepts of both functions. The \(y\)-intercept of a function is the value of \(y\) when \(x = 0\).Step 1: Find the \(y\)-intercept of \(f(x)\).The given function \(f(x) = x^2 - 2x + 12\) can be evaluated at \(x = 0\) to find its \(y\)-intercept.\(f(0) = (0)^2 - 2(0) + 12\)\(f(0) = 0 + 0 + 12\)\(f(0) = 12\)So, the \(y\)-intercept of \(f(x)\) is 12.Step 2: Determine the \(y\)-intercept of \(g(x)\) from the given details.The graph of \(g(x)\) is described as an upward-opening parabola with a vertex at (4, -7) and passing through the points (1, -3) and (7, -3). Since the \(y\)-intercept is the value of \(y\) when \(x = 0\), we need to find the equation of \(g(x)\) to determine its \(y\)-intercept.The vertex form of a parabola is given by:\(g(x) = a(x - h)^2 + k\)where \((h, k)\) is the vertex of the parabola.Given the vertex (4, -7), we can substitute \(h = 4\) and \(k = -7\) into the vertex form:\(g(x) = a(x - 4)^2 - 7\)To find the value of \(a\), we can use one of the points that the parabola passes through, such as (1, -3).\(-3 = a(1 - 4)^2 - 7\)\(-3 = a(3)^2 - 7\)\(-3 = 9a - 7\)\(9a = -3 + 7\)\(9a = 4\)\(a = \frac{4}{9}\)Now we have the equation of \(g(x)\):\(g(x) = \frac{4}{9}(x - 4)^2 - 7\)To find the \(y\)-intercept of \(g(x)\), we set \(x = 0\):\(g(0) = \frac{4}{9}(0 - 4)^2 - 7\)\(g(0) = \frac{4}{9}(16) - 7\)\(g(0) = \frac{64}{9} - 7\)\(g(0) = \frac{64}{9} - \frac{63}{9}\)\(g(0) = \frac{1}{9}\)So, the \(y\)-intercept of \(g(x)\) is \(\frac{1}{9}\).Step 3: Compare the \(y\)-intercepts of \(f(x)\) and \(g(x)\).\(f(0) = 12\)\(g(0) = \frac{1}{9}\)Since 12 is greater than \(\frac{1}{9}\), \(f\) has a greater \(y\)-intercept than \(g\).Answer:(A) \(f\) has a greater \(y\)-intercept than \(g\).