Graph and label 5 important points. 9) \( f(\theta)=4 \cos \frac{\pi}{4}(x+2)-1 \)

To graph the function \( f(\theta) = 4 \cos \left( \frac{\pi}{4} (x + 2) \right) - 1 \), we first need to analyze its components. ### Step 1: Identify the key features of the function 1. **Amplitude**: The amplitude of the cosine function is given by the coefficient in front of the cosine. Here, the amplitude is \( 4 \). 2. **Vertical Shift**: The function is shifted down by \( 1 \) unit due to the \(-1\) at the end. This means the midline of the graph is at \( y = -1 \). 3. **Period**: The period of the cosine function is calculated using the formula \( \text{Period} = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the cosine. Here, \( B = \frac{\pi}{4} \), so: \[ \text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8 \] 4. **Phase Shift**: The phase shift can be determined from the expression inside the cosine. The function can be rewritten as: \[ f(x) = 4 \cos \left( \frac{\pi}{4} x + \frac{\pi}{2} \right) - 1 \] The phase shift is given by \( -\frac{C}{B} \), where \( C \) is the constant added to \( x \). Here, \( C = 2 \), so: \[ \text{Phase Shift} = -\frac{2}{\frac{\pi}{4}} = -\frac{8}{\pi} \] ### Step 2: Important Points To find important points, we can evaluate the function at key angles for the cosine function, which are typically \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \), etc. However, since we have a phase shift and a different period, we will calculate the function at specific \( x \) values. 1. **At \( x = -2 \)**: \[ f(-2) = 4 \cos \left( \frac{\pi}{4}(-2 + 2) \right) - 1 = 4 \cos(0) - 1 = 4(1) - 1 = 3 \] Point: \( (-2, 3) \) 2. **At \( x = -2 + 2 \) (which is \( 0 \))**: \[ f(0) = 4 \cos \left( \frac{\pi}{4}(0 + 2) \right) - 1 = 4 \cos\left(\frac{\pi}{2}\right) - 1 = 4(0) - 1 = -1 \] Point: \( (0, -1) \) 3. **At \( x = -2 + 4 \) (which is \( 2 \))**: \[ f(2) = 4 \cos \left( \frac{\pi}{4}(2 + 2) \right) - 1 = 4 \cos(\pi) - 1 = 4(-1) - 1 = -5 \] Point: \( (2, -5) \) 4. **At \( x = -2 + 6 \) (which is \( 4 \))**: \[ f(4) = 4 \cos \left( \frac{\pi}{4}(4 + 2) \right) - 1 = 4 \cos\left(\frac{3\pi}{2}\right) - 1 = 4(0) - 1 = -1 \] Point: \( (4, -1) \) 5. **At \( x = -2 + 8 \) (which is \( 6 \))**: \[ f(6) = 4 \cos \left( \frac{\pi}{4}(6 + 2) \right) - 1 = 4 \cos(2\pi) - 1 = 4(1) - 1 = 3 \] Point: \( (6, 3) \) ### Summary of Important Points 1. \( (-2, 3) \) 2. \( (0, -1) \) 3. \( (2