y = sin(-3x + 3)
Amplitude =
Period =
Two x- intercepts =
y - intercept =
Domain =
Range =
Shift along x =
Shift along y =
Kind of reflection =
y = sin(-3x + 3)
sin (-a) = -sin a
so I am going to say
y = -sin (3x-3)
Amplitude = 1 implicitly
Period =
from x = 0 to 3x = 2 pi or x = (2/3) pi so period = 2 pi/3
Two x- intercepts =
y = 0 when 3x-3 = 0 or x = 1
y = 0 when 3 x - 3 = pi or x = (pi + 3)/3
y - intercept =
when x = 0, y = - sin 3
Domain = all real x
Range = -1 </= y </= +1
Shift along x = when angle is -3
Shift along y = none
Kind of reflection = do not know what this means
To find the values for each of these components in the equation y = sin(-3x + 3), we can analyze the equation and determine the necessary information.
1. Amplitude: The amplitude of a sine function represents the distance from the midline to the maximum or minimum value of the function. In this case, the amplitude is the absolute value of the coefficient in front of the sine function, which is 1. Therefore, the amplitude is 1.
2. Period: The period of a sine function is the distance between two consecutive peaks or troughs of the function. It can be calculated using the formula T = 2π/|B|, where B is the coefficient of x. In this case, B is -3, so the period is 2π/|-3| = 2π/3.
3. Two x-intercepts: The x-intercepts occur when the value of y is 0. To find the x-intercepts, set y = 0 and solve the equation for x.
sin(-3x + 3) = 0
-3x + 3 = arcsin(0)
-3x + 3 = nπ, where n is an integer
Solving for x gives us:
-3x = nπ - 3
x = (nπ - 3)/-3
Since we are interested in the two x-intercepts, we can substitute n = 0 and n = 1 to find the corresponding values of x.
When n = 0: x = (-3)/-3 = 1
When n = 1: x = (π - 3)/-3
Therefore, the two x-intercepts are x = 1 and x = (π - 3)/-3.
4. y-intercept: The y-intercept occurs when x = 0. Substitute x = 0 into the equation to find the corresponding value of y.
y = sin(-3(0) + 3) = sin(3) = 0.1411 (rounded to four decimal places)
Therefore, the y-intercept is approximately 0.1411.
5. Domain: The domain of a sine function is all real numbers since there is no restriction on the possible values of x.
Therefore, the domain is (-∞, ∞).
6. Range: The range of a sine function is between -1 and 1 since the sine function oscillates between these two values.
Therefore, the range is [-1, 1].
7. Shift along x: The shift along the x-axis occurs when the argument of the sine function, -3x + 3, is equal to 0.
-3x + 3 = 0
-3x = -3
x = 1
Therefore, there is a shift along the x-axis by 1 unit to the right.
8. Shift along y: The shift along the y-axis occurs when the constant term, in this case, 3, is added or subtracted from the function. In this equation, there is a shift of 3 units upward.
Therefore, there is a shift along the y-axis by 3 units upward.
9. Kind of reflection: The minus sign in front of the x term (-3x) causes a reflection of the graph across the x-axis.
Therefore, it is a reflection across the x-axis.