How do the graphs of f(x) = tan x and g(x) = 4 tan (x) - 7 compare? (Select all that apply)
The amplitude of g(x) is quadruple the amplitude of f(x).
g(x) anf f(x) have the same zeroes.
g(x) is shifted 7 units lower than f(x).
g(x) is shifted 4 units to the left of f(x).
clearly C
there are two answers
To compare the graphs of f(x) = tan(x) and g(x) = 4 tan(x) - 7, we can analyze each statement:
1. The amplitude of g(x) is quadruple the amplitude of f(x). - True
To determine the amplitude, we consider the highest and lowest points on the graph. For f(x) = tan(x), the highest and lowest points occur at y = 1 and y = -1, respectively, resulting in an amplitude of 1. For g(x) = 4 tan(x) - 7, the highest and lowest points occur at y = 4 + (-7) = -3 and y = -4 - (-7) = 3, respectively, resulting in an amplitude of 3 - (-3) = 6. Therefore, the amplitude of g(x) is quadruple (4 times) the amplitude of f(x).
2. g(x) and f(x) have the same zeroes. - False
The zeroes of tan(x) occur when x is an integer multiple of π (except for x = (2n+1)(π/2), where n is an integer). However, when multiplying tan(x) by 4 and subtracting 7, the zeroes will shift. Therefore, g(x) and f(x) do not have the same zeroes.
3. g(x) is shifted 7 units lower than f(x). - True
Since g(x) = 4 tan(x) - 7, the graph of g(x) will be shifted 7 units down (vertically) compared to f(x) = tan(x). This is because we are subtracting 7 from the original function.
4. g(x) is shifted 4 units to the left of f(x). - False
There is no horizontal shift mentioned in the function g(x) = 4 tan(x) - 7. Therefore, the graph of g(x) is not shifted 4 units to the left of f(x).
Hence, the correct statements are:
- The amplitude of g(x) is quadruple the amplitude of f(x).
- g(x) is shifted 7 units lower than f(x).
To understand how the graphs of f(x) = tan(x) and g(x) = 4tan(x) - 7 compare, we need to analyze each statement.
1. The amplitude of g(x) is quadruple the amplitude of f(x).
The amplitude of the tan function represents the vertical distance between its maximum and minimum values. For f(x) = tan(x), the amplitude is technically undefined since the tan function has a repeating pattern. However, we consider the range between -∞ and +∞ as the amplitude. Since the range is infinite, we can't compare the amplitudes of f(x) and g(x). Therefore, this statement is not applicable.
2. g(x) and f(x) have the same zeroes.
The zeroes of a function correspond to the x-values where the function equals zero. For f(x) = tan(x), the zeroes occur at x = nπ, where n is an integer. Similarly, for g(x) = 4tan(x) - 7, the zeroes occur when 4tan(x) - 7 = 0. Solving this equation, we get tan(x) = 7/4, which has multiple solutions, such as x = arctan(7/4) + nπ, where n is an integer. Therefore, the zeroes of g(x) and f(x) are not the same, and this statement is incorrect.
3. g(x) is shifted 7 units lower than f(x).
By examining the equation g(x) = 4tan(x) - 7, we notice that g(x) is simply the function f(x) vertically shifted downward by 7 units. This means that all points on the graph of g(x) are shifted 7 units below the corresponding points on the graph of f(x). Therefore, this statement is correct.
4. g(x) is shifted 4 units to the left of f(x).
There is no evidence or information to suggest a horizontal shift in the given equation. Both functions share the same argument, x, and have no term involving x with any coefficients other than 1. Hence, there is no horizontal shift between the graphs of f(x) and g(x). Thus, this statement is incorrect.
In summary, the correct statements are:
- g(x) is shifted 7 units lower than f(x).
- g(x) and f(x) do not have the same zeroes.