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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).

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 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.