Consider the graph of the functions \( f(x)=(x-5)^{2} \) and \( g(x)=-3 x+13 \). What are the points at which \( f(x)=g(x) \) ?
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To find the points where \( f(x) = g(x) \), we set the equations equal to each other: \[ (x-5)^{2} = -3x + 13 \] Expanding the left side: \[ x^2 - 10x + 25 = -3x + 13 \] Rearranging the equation gives: \[ x^2 - 10x + 3x + 25 - 13 = 0 \] Simplifying this results in: \[ x^2 - 7x + 12 = 0 \] Now we can factor the quadratic: \[ (x - 3)(x - 4) = 0 \] Thus, the solutions are \( x = 3 \) and \( x = 4 \). To find the corresponding \( y \)-values, we can substitute these \( x \)-values back into either function. For \( x = 3 \): \[ f(3) = (3-5)^{2} = (-2)^{2} = 4 \] For \( x = 4 \): \[ f(4) = (4-5)^{2} = (-1)^{2} = 1 \] So the points at which \( f(x) = g(x) \) are \( (3, 4) \) and \( (4, 1) \).
