Question

The graph of \(y=\cos x\) is transformed as described. Determine the values of the parameters \(a, b, c,\) and \(d\) for the transformed function. Write the equation for the transformed function in the form \(y=a \cos b(x-c)+d\). a) vertical stretch by a factor of 3 about the \(x\) -axis, horizontal stretch by a factor of 2 about the \(y\) -axis, translated 2 units to the left and 3 units up b) vertical stretch by a factor of \(\frac{1}{2}\) about the \(x\) -axis, horizontal stretch by a factor of \(\frac{1}{4}\) about the \(y\) -axis, translated 3 units to the right and 5 units down c) vertical stretch by a factor of \(\frac{3}{2}\) about the \(x\) -axis, horizontal stretch by a factor of 3 about the \(y\) -axis, reflected in the \(x\) -axis, translated \(\frac{\pi}{4}\) units to the right and 1 unit down

Short answer

a) \( y = 3 \cos( \frac{1}{2} (x + 2) ) + 3 \)b) \( y = \frac{1}{2} \cos( 4(x - 3) ) - 5 \)c) \( y = -\frac{3}{2} \cos( \frac{1}{3} (x - \frac{\pi}{4}) ) - 1 \)

Step-by-step solution

Understanding Transformations

The transformed function is given by the general form: \( y = a \cos( b(x - c) ) + d \). Here, \(a\) represents the vertical stretch or compression and reflection, \(b\) represents the horizontal stretch or compression, \(c\) represents the horizontal translation, and \(d\) represents the vertical translation.

Part a) Identifying Parameters

For the vertical stretch by a factor of 3, \(a = 3\).For the horizontal stretch by a factor of 2, \(b = \frac{1}{2}\).For the translation 2 units to the left, \(c = -2\).For the translation 3 units up, \(d = 3\). So, the equation is \( y = 3 \cos( \frac{1}{2} (x + 2) ) + 3 \).

Part b) Identifying Parameters

For the vertical stretch by a factor of \(\frac{1}{2}\), \(a = \frac{1}{2}\).For the horizontal stretch by a factor of \(\frac{1}{4}\), \(b = 4\).For the translation 3 units to the right, \(c = 3\).For the translation 5 units down, \(d = -5\). So, the equation is \( y = \frac{1}{2} \cos( 4(x - 3) ) - 5 \).

Part c) Identifying Parameters

For the vertical stretch by a factor of \(\frac{3}{2}\) and reflection in the \(x\)-axis, \(a = -\frac{3}{2}\).For the horizontal stretch by a factor of 3, \(b = \frac{1}{3}\).For the translation \( \frac{\pi}{4} \) units to the right, \(c = \frac{\pi}{4}\).For the translation 1 unit down, \(d = -1\). So, the equation is \( y = -\frac{3}{2} \cos( \frac{1}{3}(x - \frac{\pi}{4}) ) - 1 \).

Key concepts

Vertical Stretch

A vertical stretch changes the amplitude of the function. When you apply a vertical stretch by a factor of 3 to the trigonometric function \(y = \cos x\), it means that all y-values of the cosine function become three times larger. Hence, the new function would be \(y = 3 \cos x\).

For example, if the original graph of \(y = \cos x\) has points (0,1), the vertical stretch will transform it to (0,3). In general, the parameter \(a\) in the equation \(y = a \cos b(x - c) + d\) corresponds to this vertical stretch. If \(a > 1\), you have a stretch. If \(0 < a < 1\), you have a compression.

Horizontal Stretch

A horizontal stretch alters the period of the function. For instance, a horizontal stretch by a factor of 2 applied to \(y = \cos x\) makes the cosine wave stretch out and cover more of the x-axis. Mathematically, this is done by changing the parameter \(b\) in the equation \(y = a \cos b(x - c) + d\).

Specifically, if you have a stretch by a factor of 2, you modify the equation to \(y = \cos(\frac{1}{2} x)\). So every point on the graph is moved further away from the y-axis compared to the original.
For example, the period of the original \(\cos x\) function is \(2\pi\). After the stretch, it becomes \(4\pi\) since \(b = \frac{1}{2}\).

Horizontal Translation

Horizontal translation shifts the graph left or right. When you translate the cosine function by 2 units to the left, you are changing the phase of the function. This means adjusting the parameter \(c\) in the equation \( y = a \cos b(x - c) + d\).

For a translation 2 units to the left, you modify the function to \(y = \cos(x + 2)\).
The formula adopts a negative sign \( - \) to translate to the right and a positive sign \( + \) to translate to the left. In this example, the transformation translates every point on the graph of the function 2 units to the left.
For example, if you originally had the point (0,1), after a translation of 2 units left, it would be at (-2,1).

Vertical Translation

Vertical translation shifts the graph up or down. This involves changing the parameter \(d\) in the equation \( y = a \cos b(x - c) + d\). If you translate a function 3 units up, the function becomes \( y = \cos x + 3\).

Every point on the graph is moved 3 units higher than its original position. Conversely, if you translate the graph downward, \( d \) will be negative.
For instance, translating 3 units up from the point (0,1), you’d now have (0,4). So when the parameter \( d \) = 3, the entire graph of the cosine function is shifted up by 3 units.
Vertical translations affect the range of the function without altering the period or amplitude.

Reflection

Reflection flips the graph over a specific axis, usually the x-axis or y-axis. If a cosine function is reflected in the x-axis, the parameter \(a\) in the equation \( y = a \cos b(x - c) + d\) becomes negative. Thus, a reflection in the x-axis for the cosine function \( y = \cos x \) would be transformed to \(y = -\cos x\).

Every y-value of the function is inverted, turning all positive values to negative and vice versa. For instance, the point (0,1) would be reflected to (0,-1). The main use of reflections is in graph symmetry and manipulating the direction of the waves in trigonometric functions.