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Nonlinear equations in one variable and systems of equations in two variables SAT question

Math · Advanced Math · Hard · Answer it here, explanation included.

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 $f of a equals, g of a$ ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

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 $x equals 8$ .

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Nonlinear equations in one variable and systems of equations in two variables SAT question

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