How is the function f(x)=-1/2sin5x related to the function g(x)=4sin5x?
Well, it looks like f(x) and g(x) are two sinusoidal functions with the same frequency of 5x. The only difference is that f(x) has a coefficient of -1/2, while g(x) has a coefficient of 4.
We can say that f(x) is the black sheep of the family, always down in the dumps with its negative coefficient, while g(x) is the shining star, all positive and happy with its coefficient of 4. They both have the same frequency, but they differ in amplitude.
So, to summarize, f(x) is like the grumpy uncle at family gatherings, while g(x) is the cheerful cousin who's always ready to party!
To understand the relationship between the functions f(x) = -1/2sin5x and g(x) = 4sin5x, we need to consider a few factors:
1. Amplitude: The amplitude of a sine function determines the maximum distance it reaches above or below the x-axis. In the case of f(x) = -1/2sin5x, the amplitude is 1/2, meaning it reaches a maximum distance of 1/2 unit above and below the x-axis. In contrast, the amplitude of g(x) = 4sin5x is 4, indicating that it reaches a maximum distance of 4 units above and below the x-axis.
2. Vertical Stretch/Compression: The vertical stretch or compression of a sine function affects the distance between its peaks and valleys. For f(x) = -1/2sin5x, there is a vertical compression by a factor of 1/2 compared to the standard sine function. On the other hand, g(x) = 4sin5x has a vertical stretch by a factor of 4 compared to the standard sine function.
3. Phase Shift: The phase shift of a function determines the horizontal displacement of the graph. Since both f(x) = -1/2sin5x and g(x) = 4sin5x have the same expression for the argument (5x), they are both affected by the same phase shift. The phase shift depends on the value inside the argument of the sine function. In this case, it is 5. Therefore, both functions are horizontally shifted by the same amount.
In summary, the relationship between the two functions is as follows:
- The amplitude of f(x) is smaller than the amplitude of g(x).
- f(x) is vertically compressed compared to g(x).
- Both functions undergo the same horizontal shift.
To understand how the functions f(x) = -1/2sin(5x) and g(x) = 4sin(5x) are related, let's break down each function separately and then compare them.
1. Function f(x) = -1/2sin(5x):
This function involves the sine function, where 5x represents the argument of the sine function. The negative sign in front of the half (-1/2) indicates that the graph is reflected about the x-axis, resulting in an upside-down graph compared to the standard sine function. The coefficient of 5 (5x) determines the frequency of the graph, causing it to oscillate more quickly. Overall, function f(x) decreases the amplitude of the standard sine function by a factor of 1/2 and reflects it vertically.
2. Function g(x) = 4sin(5x):
Similar to function f(x), this function also involves the sine function with an argument of 5x. However, the coefficient of 4 increases the amplitude of the standard sine function by a factor of 4. Hence, function g(x) results in a graph that oscillates with a larger amplitude compared to the standard sine function.
Now, let's compare the two functions:
The only difference between the two functions f(x) and g(x) is the coefficient in front of the sine function. While function f(x) has a coefficient of -1/2, function g(x) has a coefficient of 4. This means that function g(x) has a larger amplitude compared to function f(x) since the coefficient of 4 increases the oscillation. Moreover, both functions have the same frequency, given by the argument 5x.
In summary, the functions f(x) = -1/2sin(5x) and g(x) = 4sin(5x) have the same frequency of oscillation given by 5x. However, the coefficient affects the amplitude of the graphs, where f(x) has a smaller amplitude and g(x) has a larger amplitude compared to the standard sine function.
f(x) = a * g(x)
(-1/2) sin 5x = a * 4 sin 5x
4 a = -1/2
a = -1/8
so
f(x) = (-1/8) g(x)