OG详解-OG6 数学1 Q26

Choice A is correct. It’s given that the table shows values of x and their corresponding values of g(x), where g(x) = f(x)x+3​​. It’s also given that f is a linear function. It follows that an equation that defines f can be written in the form f(x) = mx+b, where m represents the slope and b represents the y-coordinate of the y-intercept (0, b) of the graph of y = f(x) in the xy-plane. The slope of the graph of y = f(x) can be found using two points, (x₁, y₁) and (x₂, y₂), that are on the graph of y = f(x), and the formula m = y₂−y1x2−x1​​. Since the table shows values of x and their corresponding values of g(x), substituting values of x and g(x) in the equation g(x) =f(x)x+3​​ can be used to define function f. Using the first pair of values from the table, x = -27 and g(x) = 3, yields 3 = f(−27)27+3​​  , or 3 = f(−27)−24​​. Multiplying each side of this equation by -24 yields -72 = f(-27), so the point (-27, -72) is on the graph of y = f(x). Using the second pair of values from the table, x = -9 and g(x) = 0, yields 0 = f(−9)−9+3​​ , or 0 = f(−9)−6​​. Multiplying each side of this equation by -6 yields 0 = f(-9), so the point (-9, 0) is on the graph of y = f(x). Substituting (-27, -72) and (-9, 0) for (x1, y1) and (x2, y2), respectively, in the formula m = y2−y1x2−x1​​  yields m = 0−(−72)−9−(−27​)​ , or m = 4. Substituting 4 for m in the equation f(x) = mx+b yields f(x) = 4x+b. Since 0 = f(-9), substituting -9 for x and 0 for f(x) in the equation f(x) = 4x+b yields 0 = 4(-9)+b, or 0 =-36+b. Adding 36 to both sides of this equation yields 36 = b. It follows that 36 is the y-coordinate of the y-intercept (0, b) of the graph of y = f(x). Therefore,  the y-intercept of the graph of y = f(x) is (0, 36).
Choice B is incorrect. 12 is the y-coordinate of the y-intercept of the graph of y = g(x). Choice C is incorrect. 4 is the slope of the graph of y = f(x). Choice D is incorrect. -9 is the x-coordinate of the x-intercept of the graph of y = f(x).