A curve is described by f(x) = x^2 + 2x. A second curve is described by g(x) = x^2 -5x + 7. Find the point (s) where both curves intersect.
To find the points where two curves meet a difference function needs to be calculated. This function is formed by subtracting one function from the other: d(x) = f(x) - g(x). It also works the other way around: d'(x) = g(x) - f(x). The reason we did this is to determine the points where the two functions meet. At these points both functions have the same exact value. Consequently as the value for g(x) is equal to f(x) at these points the difference function d(x) is zero. Now to determine these points we need to set d(x) = 0 and calculate these points:d(x) = f(x)- g(x) = x2 +2x - (x2 -5x + 7) = 7x - 7At f(x) = g(x) (the points where f and g meet) d(x) = 0:d(x) = 0 => 7x - 7 = 0 <=> 7x = 7 <=> x=1That means that at the x = 1 position both curves meet. Now we need to calculate the y-value.For this we plug our determined x value into f or g:f(1) = 12 + 21 = 3 This means at the point (1 | 3) both functions meet. To double check our calculation we can calculate the value of g(1):g(1) = 12 - 51 + 7 = 3Having double checked our answer we can be certain that it is correct.
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