Draw this graph [0;270] -1/2tan+3/4 And find the domain and the range and period

Here's the graph of y = -(1)/(2)(x) + (3)/(4) for the interval [0, 270^]. Step 1: Identify the transformations. The graph is a transformation of the basic tangent function y = (x). The -(1)/(2) factor vertically compresses the graph and reflects it across the x-axis. The +(3)/(4) shifts the graph vertically upwards by (3)/(4) units. Step 2: Determine the period of the function. The period of y = a (bx + c) + d is ()/(|b|) (in radians) or (180^)/(|b|) (in degrees). In this function, y = -(1)/(2)(x) + (3)/(4), b=1. So, the period is (180^)/(1) = 180^. Step 3: Find the vertical asymptotes. The basic tangent function y = (x) has vertical asymptotes at x = 90^ + 180^ n, where n is an integer. For y = -(1)/(2)(x) + (3)/(4), the vertical asymptotes remain at x = 90^ + 180^ n. Within the interval [0^, 270^], the asymptotes are at x = 90^ and x = 90^ + 180^ = 270^. Step 4: Find the x-intercept (or where the graph crosses the horizontal shift line). The horizontal shift line is y = (3)/(4). We set -(1)/(2)(x) + (3)/(4) = (3)/(4). -(1)/(2)(x) = 0 (x) = 0 This occurs at x = 0^, 180^, 360^, . Within the interval [0^, 270^], these points are x = 0^ and x = 180^. Step 5: Evaluate the function at key points. At x = 0^: y = -(1)/(2)(0^) + (3)/(4) = -(1)/(2)(0) + (3)/(4) = (3)/(4). Point: (0^, (3)/(4)). At x = 180^: y = -(1)/(2)(180^) + (3)/(4) = -(1)/(2)(0) + (3)/(4) = (3)/(4). Point: (180^, (3)/(4)). As x approaches 90^ from the left (x 90^-), (x) +, so -(1)/(2)(x) -, and y -. As x approaches 90^ from the right (x 90^+), (x) -, so -(1)/(2)(x) +, and y +. As x approaches 270^ from the left (x 270^-), (x) +, so -(1)/(2)(x) -, and y -. Step 6: Sketch the graph. Plot the points (0^, (3)/(4)) and (180^, (3)/(4)). Draw vertical asymptotes at x = 90^ and x = 270^. Sketch the curve, remembering that the negative coefficient flips the usual upward slope of the tangent function. The graph will decrease from left to right between asymptotes. Graph of y = -(1)/(2)(x) + (3)/(4) for [0^, 270^] (Imagine a graph here with vertical asymptotes at x=90^ and x=270^. The curve passes through (0^, 0.75) and (180^, 0.75). Between 0^ and 90^, the curve goes down from 0.75 towards -. Between 90^ and 180^, the curve comes from + and goes down to 0.75. Between 180^ and 270^, the curve goes down from 0.75 towards -.) Step 7: Find the domain. The domain is restricted by the vertical asymptotes. The function is undefined at x = 90^ and x = 270^. The given interval is [0^, 270^]. So, the domain is [0^, 90^) (90^, 270^). Step 8: Find the range. The tangent function, even with transformations, covers all real numbers. The vertical shift and compression/reflection do not change this. The range is (-, ). The period is 180^. The domain is [0^, 90^) (90^, 270^]. The range is (-, ). The period is 180^. Got more? Send 'em 📸