State domain, range, period, vertical asymptotes, zeros, symmetry and y-intercept of y = -2tan(3x + 180°) + 3

I just need to know if there is a formula to find the vertical asymptotes, symmetry, and zeros or do I have to solve it graphically.

Thanks.

You can get the vertical asymptote by solving dy/dx = infinity

You can get the y intercept by solving
y = f(0)= 2 tan(180) + 3 = 3

To find the domain, range, period, vertical asymptotes, zeros, symmetry, and y-intercept of the given function y = -2tan(3x + 180°) + 3, we can use various mathematical techniques. Let's break it down step by step:

1. Domain: The domain of the function is the set of all possible x-values for which the function is defined. In this case, the tangent function is undefined for values where the angle (3x + 180°) results in ±nπ/2 (where n is an integer). Therefore, the domain is all real numbers except for those that make (3x + 180°) equal to (±nπ/2). To find the precise domain, solve the equation (3x + 180°) = (±nπ/2) for x.

2. Range: The range of the function is the set of all possible y-values that the function can produce. For the tangent function, the range is all real numbers, including positive and negative infinity. In this case, since the -2 multiplier is present, it shifts the graph vertically, which means the range will be limited. By analyzing the -2 multiplier and the +3 constant term, you can conclude that the range will be between positive infinity and 3, inclusive.

3. Period: The period of the function is the horizontal length of one complete cycle of the graph. For the generic function y = A tan(Bx + C), the period is given by the formula 2π/B. In this case, the period will be 2π/3 because the coefficient of x is 3.

4. Vertical Asymptotes: The vertical asymptotes of the function occur at x-values that make the tangent function undefined. As mentioned earlier, these x-values occur when (3x + 180°) = (±nπ/2). By solving this equation for x, you can find the vertical asymptotes.

5. Zeros: The zeros (also known as x-intercepts) of the function occur when y = 0. In this case, solve the equation -2tan(3x + 180°) + 3 = 0 for x to find the zeros.

6. Symmetry: To determine the symmetry, you need to determine if the function is odd, even, or neither. In this case, the function is neither odd nor even because the presence of the tangent function does not exhibit either symmetry property.

7. Y-intercept: The y-intercept occurs when x = 0. Substitute x = 0 into the function to find the y-intercept. In this case, substitute x = 0 and solve the equation to find the y-intercept.

As for finding these values precisely, solving equations and graphing techniques can be used. Some calculations might be complex, but using a graphing calculator or mathematical software can simplify the process.