Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=5 \cos (\pi x)-3 $$

Short answer

Domain: \((\infty, +\infty)\), Range: \([-8, 2]\)

Step-by-step solution

- Identify the basic function

The basic function here is the cosine function: \(y = \cos(x)\). The function \(y = 5 \cos (\pi x) - 3\) represents a transformed version of this basic cosine function.

- Identify the amplitude

The coefficient '5' in front of the cosine affects the amplitude. Thus, the amplitude of the function \(y = 5 \cos (\pi x) - 3\) is 5.

- Identify the period

The coefficient '\(\pi\)' inside the cosine function affects the period. The period of the basic cosine function is \(2\pi\). For \(y = 5 \cos (\pi x) - 3\), the period is calculated as: \[ \text{Period} = \frac{2\pi}{\pi} = 2 \]

- Identify the vertical shift

The '-3' at the end of the function shifts the entire graph down by 3 units. So, the vertical shift is -3.

- Identify the key points

Determine the key points. For the function \(y = 5 \cos(\pi x) - 3\), the key points for one period, starting at \(x=0\), are: \[ \begin{align*} f(0) &= 5 \cos(0) - 3 = 2, \ f(\frac{1}{2}) &= 5 \cos(\frac{\pi}{2}) - 3 = -3, \ f(1) &= 5 \cos(\pi) - 3 = -8, \ f(\frac{3}{2}) &= 5 \cos(\frac{3\pi}{2}) - 3 = -3, \ f(2) &= 5 \cos(2\pi) - 3 = 2. \end{align*} \]

- Graph the function

Plot the key points as determined in Step 5: \[(0, 2), (\frac{1}{2}, -3), (1, -8), (\frac{3}{2}, -3), (2, 2)\]. Repeat for at least two cycles and draw a smooth curve through these points.

- Determine the domain

The domain of the function \(y = 5 \cos (\pi x) - 3\) is all real numbers, so the domain is \((\infty, +\infty)\).

- Determine the range

The range of the function is determined by the maximum and minimum values. Since the amplitude is 5 and the vertical shift is -3, the function oscillates between \[2 = 5 - 3\] and \[-8 = -5 - 3\]. Therefore, the range is \([-8, 2]\).

Key concepts

function transformations

Function transformations involve changes to the parent function that move or scale the graph. For the given function, we start with the basic cosine function, which is transformed in several ways:

• The amplitude change, controlled by the coefficient in front of the cosine.
• The period change, controlled by the coefficient inside the cosine function.
• The vertical shift, controlled by the constant added or subtracted at the end.

By analyzing these transformations, we can effectively graph the transformed function and understand its behavior.

amplitude

Amplitude refers to the height of the peaks and the depth of the troughs from the midline of the function. For the function
\( y = 5 \cos(\pi x) - 3 \),
the coefficient 5 in front of the cosine affects the amplitude. This means our amplitude is 5, so the graph will oscillate 5 units above and below the midline (horizontal axis).
The amplitude is crucial as it tells us how far the function strays from its central axis.

period

The period of a trigonometric function refers to the length of one complete cycle of the wave. For the function\( y = 5 \cos(\pi x) - 3 \):
the coefficient \(\pi\) inside the cosine function influences the period. The formula to determine the period of a cosine function is:
\[ \text{Period} = \frac{2\pi}{\text{coefficient of x in } \cos} = \frac{2\pi}{\pi} = 2 \]
Therefore, the function completes one cycle every 2 units along the x-axis.

vertical shift

The vertical shift moves a function up or down along the y-axis. In our function,
\( y = 5 \cos(\pi x) - 3 \),
the term \(-3\) at the end shifts the entire graph downward by 3 units. This means every point on the basic cosine curve is moved 3 units lower. Vertical shifts are important for adjusting the central axis of the wave to better fit the problem's context.

domain and range

The domain specifies all the possible input (x) values, while the range specifies all the possible output (y) values that a function can take.
For the cosine function
\( y = 5 \cos(\pi x) - 3 \),

• The domain is all real numbers because you can input any real number into the cosine function.
• This can be written as \((\infty,+\infty)\).

The range is determined by the amplitude and vertical shift. As calculated, the function oscillates between:
\[ [-8, 2] \]
Where -8 is the minimum value (\(-5\) from the amplitude minus \(3\) from the vertical shift) and 2 is the maximum value (\(\ 5\) from the amplitude minus +\(3\) from the vertical shift).