Problem 32 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=\cos x\) by basic transformations: (a) \(y=1+2 \cos x\) (b) \(y=-\cos \left(x+\frac{\pi}{4}\right)\) (c) \(y=-\cos \left(\frac{\pi}{2}-x\right)\)

Step-by-Step Solution

Answer

(a) Stretch by 2 and shift up by 1; (b) Reflect and shift left by \(\frac{\pi}{4}\); (c) Convert using angle identity and reflect.

Step 1: Identify Vertical Stretch and Shift for (a)

The function given is \( y = 1 + 2 \cos x \). Start by identifying the transformations applied to \( y = \cos x \). Here, the \( 2 \) represents a vertical stretch by a factor of 2. The \( +1 \) indicates a vertical shift upwards by 1 unit. Therefore, you multiply the cosine function by 2 and then add 1 to the result.

Step 2: Determine Reflections and Phase Shift for (b)

For \( y = -\cos \left(x+\frac{\pi}{4}\right) \), the negative sign indicates a reflection over the x-axis. The term \( x+\frac{\pi}{4} \) indicates a phase shift to the left by \( \frac{\pi}{4} \). Thus, the original cosine function is reflected and then shifted left.

Step 3: Identify Complementary Angle Identity for (c)

Rewrite \( y = -\cos \left(\frac{\pi}{2}-x\right) \) using the complementary angle identity. Recall that \( \cos(\frac{\pi}{2}-x) = \sin(x) \), so the function becomes \( y = -\sin x \). Thus, \( y = \cos x \) is transformed by first converting \( \cos \) to \( \sin \) (due to the identity) and then reflecting over the x-axis due to the negative sign.

Key Concepts

Vertical Stretch and Shift

A vertical stretch changes the amplitude of a function, making it taller or shorter. When you multiply a function by a number greater than 1, the graph stretches vertically by that factor. In the given function, \[ y = 1 + 2 \cos x \] The number 2 causes this vertical stretch. It multiplies the original cosine values, making the peaks and troughs twice as far from the x-axis as in the original function, \( y = \cos x \).Additionally, a vertical shift is when the entire graph moves up or down without changing its shape. Here, the "\(+1\)" indicates that the whole function is moved upward by 1 unit. This signifies that every point on the graph of \( y = 2\cos x \) is lifted up one unit on the y-axis.

Quick summary:

• Multiply the amplitude (vertical stretch) by 2.
• Shift the graph up by 1 unit (vertical shift).

Phase Shift

A phase shift refers to moving the graph of a function left or right. In the expression for horizontal shifts within a trigonometric function, \[ y = -\cos \left(x+\frac{\pi}{4}\right) \]The term inside the cosine function, "\(x + \frac{\pi}{4}\)", highlights a shift in the opposite direction of the sign. Since it is a "\(+\)" sign, it causes a leftward shift. Moving left by \(\frac{\pi}{4}\) means that features like peaks, troughs, and zero-crossings occur earlier on the graph.

There is also a reflection over the x-axis which flips the graph upside down. This is indicated by the negative sign in front of the cosine function. Together:

• Reflect the graph over the x-axis.
• Shift the entire graph to the left by \(\frac{\pi}{4}\).

Reflection Over X-Axis

Reflection over the x-axis results in flipping the graph. For trigonometric functions, a negative sign in front of the function causes any positive value to become negative and vice versa. Consider the transformation:\[ y = -\cos \left(\frac{\pi}{2}-x\right) \]Initially, you can use the complementary angle identity, \( \cos(\frac{\pi}{2}-x) = \sin(x) \), to rewrite it as:\[ y = -\sin x \]Here, the negative sign indicates a reflection of \( \sin x \) over the x-axis. This flips all the points on the graph as if you’re looking at it in a mirror held along the x-axis. What was above the x-axis is now below, and what was below is now above.

Key points about reflection:

• Negative sign means the function is upside down.
• It affects both cosine and sine functions the same way.