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Algebra / Linear functions Difficulty: Hard

x	$negative 11$	$negative 10$	$negative 9$	$negative 8$
f(x)	21	18	15	12

The table above shows some values of x and their corresponding values $f of x$ for the linear function f. What is the x-intercept of the graph of $y equals f of x$ in the xy-plane?

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Explanation

Choice B is correct. The equation of a linear function can be written in the form $y equals, m x plus b$ , where $y equals f of x$ , m is the slope of the graph of $y equals f of x$ , and b is the y-coordinate of the y-intercept of the graph. The value of m can be found using the slope formula, $m equals the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction$ . According to the table, the points $with coordinates negative 11 comma 21$ and $with coordinates negative 10 comma 18$ lie on the graph of $y equals f of x$ . Using these two points in the slope formula yields $m equals the fraction with numerator 18 minus 21, and denominator negative 10 plus 11, end fraction$ , or $negative 3$ . Substituting $negative 3$ for m in the slope-intercept form of the equation yields $y equals, negative 3 x plus b$ . The value of b can be found by substituting values from the table and solving; for example, substituting the coordinates of the point $with coordinates negative 11 comma 21$ into the equation $y equals, negative 3 x plus b$ gives $21 equals, negative 3 times negative 11, plus b$ , which yields $b equals negative 12$ . This means the function given by the table can be represented by the equation $y equals, negative 3 x minus 12$ . The value of the x-intercept of the graph of $y equals f of x$ can be determined by finding the value of x when $y equals 0$ . Substituting $y equals 0$ into $y equals, negative 3 x minus 12$ yields $0 equals, negative 3 x minus 12$ , or $x equals negative 4$ . This corresponds to the point $with coordinates negative 4 comma 0$ .

Choice A is incorrect and may result from substituting the value of the slope for the x-coordinate of the x-intercept. Choice C is incorrect and may result from a calculation error. Choice D is incorrect and may result from substituting the y-coordinate of the y-intercept for the x-coordinate of the x-intercept.