In the diagram below, the graphs of f(x) = a/x + q and g(x) = -x + 4 are drawn. Graphs of f and g intersect at A and C. f and g intersect the x-axis at point E and D respectively. B lies on the y-axis. The horizontal asymptote of f intersect g at point B.

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Asked on June 6, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-06-06T15:44:34.956Z

Step 1: Determine the coordinates of B. Point B lies on the y-axis, so its x-coordinate is 0. Point B is also the intersection of the horizontal asymptote of f and the graph of g. The equation of g(x) is given as g(x) = -x + 4. Substitute x=0 into g(x) to find the y-coordinate of B: g(0) = -(0) + 4 = 4 So, the coordinates of B are (0, 4). Since B is on the horizontal asymptote of f(x) = (a)/(x) + q, and the horizontal asymptote is y=q, we can conclude that q=4. The coordinates of B are (0, 4). Step 2: Determine the equation of f. The general equation for f is f(x) = (a)/(x) + q. From Step 1, we found q=4. So, f(x) = (a)/(x) + 4. We are given that point P((1)/(4); -4) lies on the graph of f. Substitute these coordinates into the equation for f: -4 = (a)/(1)4 + 4 -4 = 4a + 4 Subtract 4 from both sides: -4 - 4 = 4a -8 = 4a Divide by 4: a = (-8)/(4) a = -2 Therefore, the equation of f is f(x) = -(2)/(x) + 4. The equation of f is f(x) = -(2)/(x) + 4. Step 3: Determine the x-coordinates of A and C. Points A and C are the intersection points of f(x) and g(x). Set f(x) = g(x): -(2)/(x) + 4 = -x + 4 Subtract 4 from both sides: -(2)/(x) = -x Multiply both sides by -x (assuming x ≠ 0): 2 = x^2 Take the square root of both sides: x = ± sqrt(2) From the graph, point A is in the second quadrant (negative x-value) and point C is in the first quadrant (positive x-value). So, the x-coordinate of A is -sqrt(2) and the x-coordinate of C is sqrt(2). The x-coordinates of A and C are x_A = -sqrt(2) and x_C = sqrt(2). Step 4: Calculate the area of BOD. To calculate the area of BOD, we need the coordinates of B, O, and D. O is the origin, so its coordinates are (0, 0). From Step 1, the coordinates of B are (0, 4). D is the x-intercept of g(x). To find D, set g(x) = 0: -x + 4 = 0 x = 4 So, the coordinates of D are (4, 0). BOD is a right-angled triangle with the right angle at O. The length of the base OD is the distance from (0,0) to (4,0), which is 4 units. The length of the height OB is the distance from (0,0) to (0,4), which is 4 units. The area of a triangle is given by the formula: Area = (1)/(2) × base × height. Area of BOD = (1)/(2) × OD × OB Area of BOD = (1)/(2) × 4 × 4 Area of BOD = (1)/(2) × 16 Area of BOD = 8 square units The area of BOD is 8 square units. Send me the next one 📸

Accepted Answer · 2026-06-06T15:44:34.956Z

Step 1: Determine the coordinates of B. Point B lies on the y-axis, so its x-coordinate is 0. Point B is also the intersection of the horizontal asymptote of f and the graph of g. The equation of g(x) is given as g(x) = -x + 4. Substitute x=0 into g(x) to find the y-coordinate of B: g(0) = -(0) + 4 = 4 So, the coordinates of B are (0, 4). Since B is on the horizontal asymptote of f(x) = (a)/(x) + q, and the horizontal asymptote is y=q, we can conclude that q=4. The coordinates of B are (0, 4). Step 2: Determine the equation of f. The general equation for f is f(x) = (a)/(x) + q. From Step 1, we found q=4. So, f(x) = (a)/(x) + 4. We are given that point P((1)/(4); -4) lies on the graph of f. Substitute these coordinates into the equation for f: -4 = (a)/(1)4 + 4 -4 = 4a + 4 Subtract 4 from both sides: -4 - 4 = 4a -8 = 4a Divide by 4: a = (-8)/(4) a = -2 Therefore, the equation of f is f(x) = -(2)/(x) + 4. The equation of f is f(x) = -(2)/(x) + 4. Step 3: Determine the x-coordinates of A and C. Points A and C are the intersection points of f(x) and g(x). Set f(x) = g(x): -(2)/(x) + 4 = -x + 4 Subtract 4 from both sides: -(2)/(x) = -x Multiply both sides by -x (assuming x ≠ 0): 2 = x^2 Take the square root of both sides: x = ± sqrt(2) From the graph, point A is in the second quadrant (negative x-value) and point C is in the first quadrant (positive x-value). So, the x-coordinate of A is -sqrt(2) and the x-coordinate of C is sqrt(2). The x-coordinates of A and C are x_A = -sqrt(2) and x_C = sqrt(2). Step 4: Calculate the area of BOD. To calculate the area of BOD, we need the coordinates of B, O, and D. O is the origin, so its coordinates are (0, 0). From Step 1, the coordinates of B are (0, 4). D is the x-intercept of g(x). To find D, set g(x) = 0: -x + 4 = 0 x = 4 So, the coordinates of D are (4, 0). BOD is a right-angled triangle with the right angle at O. The length of the base OD is the distance from (0,0) to (4,0), which is 4 units. The length of the height OB is the distance from (0,0) to (0,4), which is 4 units. The area of a triangle is given by the formula: Area = (1)/(2) × base × height. Area of BOD = (1)/(2) × OD × OB Area of BOD = (1)/(2) × 4 × 4 Area of BOD = (1)/(2) × 16 Area of BOD = 8 square units The area of BOD is 8 square units. Send me the next one 📸

In the diagram below, the graphs of f(x) = a/x + q and g(x) = -x + 4 are drawn. Graphs of f and g intersect at A and C. f and g intersect the x-axis at point E and D respectively. B lies on the y-axis. The horizontal asymptote of f intersect g at point B.

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In the diagram below, the graphs of f(x) = a/x + q and g(x) = -x + 4 are drawn. Graphs of f and g intersect at A and C. f and g intersect the x-axis at point E and D respectively. B lies on the y-axis. The horizontal asymptote of f intersect g at point B.

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