Modded SAT Question Bank
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Test
Math
Domain
Advanced Math
Skill
Nonlinear equations in one variable and systems of equations in two variables
Difficulty
Hard
ID: 97e50fa2
Modded SAT Question Bank by Abdullah Mallik
The figure presents the graph of a curve in the xy-plane. The curve is labeled y equals f of x. The numbers 0 through 9 are indicated on the x-axis. The numbers 0 through 12, in increments of 2, are indicated on the y-axis. The curve begins at the point with coordinates 0 comma 2 and moves upward and to the right reaching a maximum at the point with coordinates 4 comma 10. It turns and moves downward and to the right, ending at 8 point 5 on the x-axis

The graph of the function f, defined by f of x equals, negative one-half times, open parenthesis, x minus 4, close parenthesis, squared, plus 10, is shown in the xy-plane above. If the function g (not shown) is defined by g of x equals, negative x plus 10, what is one possible value of a such that f of a equals, g of a ?


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Correct Answer:
Rationale

The correct answer is either 2 or 8. Substituting x equals a in the definitions for f and g gives f of a, equals, negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, plus 10 and g of a, equals, negative a, plus 10, respectively. If f of a, equals, g of a, then negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, plus 10, equals, negative a, plus 10. Subtracting 10 from both sides of this equation gives negative one half times, open parenthesis, a, minus 4, close parenthesis, squared, equals negative a. Multiplying both sides by negative 2 gives open parenthesis, a, minus 4, close parenthesis, squared, equals 2 a. Expanding open parenthesis, a, minus 4, close parenthesis, squared gives a, squared, minus 8 a, plus 16, equals 2 a. Combining the like terms on one side of the equation gives a, squared, minus 10 a, plus 16, equals 0. One way to solve this equation is to factor a, squared, minus 10 a, plus 16 by identifying two numbers with a sum of negative 10 and a product of 16. These numbers are negative 2 and negative 8, so the quadratic equation can be factored as open parenthesis, a, minus 2, close parenthesis, times, open parenthesis, a, minus 8, close parenthesis, equals 0. Therefore, the possible values of a are either 2 or 8. Note that 2 and 8 are examples of ways to enter a correct answer.

Alternate approach: Graphically, the condition f of a, equals, g of a implies the graphs of the functions y equals f of x and y equals g of x intersect at x equals a. The graph y equals f of x is given, and the graph of y equals g of x may be sketched as a line with y-intercept 10 and a slope of negative 1 (taking care to note the different scales on each axis). These two graphs intersect at x equals 2 and x equals 8.

Question Difficulty: Hard
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