y = cos(4x/3 minus 1)
amplitude?
Period?
Two x - intercepts?
y-intercept?
Domain?
Range?
Shift along x?
Shift along y?
Kind of reflection?
amplitude=1 since this is the maximum value of the function.
Period is value of x when:
4x/3 minus 1 = 2pi.
x intercepts are when:
4x/3 minus 1 = pi/2
4x/3 minus 1 = pi+(pi/2)
y intercept is when
4x/3 minus 1 = 0
Domain is minus infinity to plus infinity.
Range is -1 to +1
To find the amplitude, period, x-intercepts, y-intercept, domain, range, shift along x, shift along y, and kind of reflection for the given function y = cos(4x/3 - 1), we can analyze the equation.
1. Amplitude:
The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine term. In this case, the coefficient is 1, so the amplitude is 1.
2. Period:
The period of a cosine function is given by the formula: 2π/|B|, where B is the coefficient of x inside the cosine function. In this case, the coefficient is 4/3, so the period is 2π/(4/3) = 3π/2.
3. Two x-intercepts:
To find the x-intercepts, we set y = 0 and solve for x. In this case, we have:
cos(4x/3 - 1) = 0
4x/3 - 1 = π/2 + kπ
4x/3 = π/2 + kπ + 3/2 for integer values of k
x = 3/4(π/2 + kπ + 3/2), where k is an integer
So there are infinitely many x-intercepts, spaced π/2 apart, starting from x = -3/4.
4. Y-intercept:
The y-intercept is the value of the function when x = 0. Plugging in x = 0 into the equation:
y = cos(4(0)/3 - 1) = cos(-1) ≈ 0.5403.
5. Domain:
The domain is the set of all possible x-values for which the function is defined. Since cosine is defined for all real numbers, the domain of this function is (-∞, ∞).
6. Range:
The range is the set of all possible y-values that the function can take. The cosine function has a range of [-1, 1]. However, since there is a shift along the y-axis, the range in this case is [y-shift - amplitude, y-shift + amplitude]. So the range is approximately [-1 - (y-shift), 1 + (y-shift)].
7. Shift along x:
The function has a shift along x due to the term in the argument of the cosine function. The shift is determined by the equation Bx - C = 0, where B is the coefficient of x and C is the constant term. In this case, B = 4/3 and C = 1. So the shift along x is 3/4.
8. Shift along y:
The function has a shift along y due to the constant term outside the cosine function. In this case, the shift along y is -1.
9. Kind of reflection:
The function reflects the graph of the cosine function across the x-axis due to the negative sign in front of the argument of the cosine function inside the parentheses.
I hope this explanation helps you understand how to find the different characteristics of the function y = cos(4x/3 - 1).