The graph of the function f, defined by `f(x)=-1/2(x-4)^2 +10`, is shown in the xy-plane above. If the function g (not shown) is defined by g(x) = −x + 10, what is one possible value of a such that f (a) = g(a) ?

Answer

2 AND 8

Explanation

Substituting x = a in the definitions for f and g gives `f(a)=-1/2(a-4)^2 + 10` and g(a)=-a+10, respectively. If f(a)=g(a) , then `-1/2(a-4)^2+10 =-a +10` Subtracting 10 from both sides of this equation gives `-1/2(a-4)^2=-a` . Multiplying both sides by -2 gives `(a-4)^2=2a`. Expanding `(a-4)^2` gives `a^2-8a+16=2a`. Combining the like terms on one side of the equation gives `a^2 − 10a + 16 = 0`. One way to solve this equation is to factor `a^2 − 10a + 16` by identifying two numbers with a sum of −10 and a product of 16. These numbers are −2 and −8, so the quadratic equation can be factored as (a − 2)(a − 8) = 0. Therefore, the possible values of a are either 2 or 8. Either 2 or 8 will be scored as a correct answer.
Alternate approach: Graphically, the condition f (a) = g(a) implies the graphs of the functions y = f (x) and y = g(x) intersect at x = a. The graph y = f (x) is given, and the graph of y = g(x) may be sketched as a line with y-intercept 10 and a slope of −1 (taking care to note the different scales on each axis). These two graphs intersect at x = 2 and x = 8.