To analyze and describe the given function [tex]\( f(x) = 2 \sin 4 (x - 45^{\circ}) + 1 \)[/tex]:
1. Amplitude:
The amplitude of a trigonometric function [tex]\( a \sin(bx+c) + d \)[/tex] or [tex]\( a \cos(bx+c) + d \)[/tex], is given by the coefficient [tex]\( |a| \)[/tex]. Here, for [tex]\( f(x) = 2 \sin 4 (x - 45^{\circ}) + 1 \)[/tex], [tex]\( a = 2 \)[/tex]. Therefore, the amplitude of the function is:
[tex]\[
\text{Amplitude} = 2
\][/tex]
2. Midline Equation:
The midline of the function is determined by the vertical shift, which is the constant term added to the sine function. Here, the constant term is [tex]\( +1 \)[/tex], so the midline of the function is:
[tex]\[
\text{Midline: } y = 1
\][/tex]
3. Range:
The range of the function is determined by how much the function oscillates above and below the midline. Since the amplitude is 2, the function will reach a maximum value of [tex]\( 1 + 2 = 3 \)[/tex] and a minimum value of [tex]\( 1 - 2 = -1 \)[/tex]. Hence, the range of the function is:
[tex]\[
\text{Range: } [-1, 3]
\][/tex]
4. Period:
The period of a sine function [tex]\( a \sin(bx + c) + d \)[/tex] is given by [tex]\( \frac{360^{\circ}}{b} \)[/tex], where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( b = 4 \)[/tex], so the period of the function is:
[tex]\[
\text{Period: } \frac{360^{\circ}}{4} = 90^{\circ}
\][/tex]
5. Horizontal Translation:
The horizontal translation (also known as phase shift) is determined by the expression inside the sine function [tex]\( 4(x - 45^{\circ}) \)[/tex]. The phase shift is given by solving [tex]\( x - \text{Shift} = 0 \)[/tex]. From [tex]\( 4(x - 45^{\circ}) \)[/tex], the shift is [tex]\( +45^{\circ} \)[/tex], indicating a shift to the right by 45 degrees. Hence, the horizontal translation is:
[tex]\[
\text{Horizontal Translation: } 45^{\circ} \text{ to the right}
\][/tex]
In summary, for the function [tex]\( f(x) = 2 \sin 4 (x - 45^{\circ}) + 1 \)[/tex]:
- Amplitude: 2
- Midline Equation: [tex]\( y = 1 \)[/tex]
- Range: [tex]\([-1, 3]\)[/tex]
- Period: [tex]\( 90^{\circ} \)[/tex]
- Horizontal Translation: [tex]\( 45^{\circ} \)[/tex] to the right
