Find the equation for each curve in its final position. The graph of y=cos(x) is stretched by a

Find the equation for each curve in its final position. The...

Key Concepts

Function Transformations

Function transformations are operations that modify the appearance of a graph. They include operations such as stretching, shifting, translating, and reflecting. Understanding these changes allows one to predict how alterations to a function’s formula affect its graph, an essential skill in algebra and precalculus.

Vertical Stretch

A vertical stretch involves multiplying the output (y-values) of a function by a factor greater than one. This transformation increases the distance of each point from the x-axis without affecting the x-coordinates, thereby making the graph appear taller or more 'stretched' in the vertical direction.

Horizontal Translation

A horizontal translation shifts a graph left or right without changing its shape. This is done by adding or subtracting a constant within the function’s argument. The operation moves every point on the graph by the same distance in the horizontal direction, altering its position along the x-axis.

Vertical Translation

A vertical translation moves the graph up or down on the coordinate plane. This is achieved by adding or subtracting a constant to the entire function. The process shifts the entire graph uniformly in the vertical direction without affecting its shape.

Reflection in the x-axis

Reflection in the x-axis inverts the graph across the x-axis. Mathematically, this is accomplished by multiplying the entire function by -1. This transformation changes the sign of all the y-values, creating a mirror image of the original graph across the horizontal axis.

Composite Transformations

Composite transformations combine multiple individual transformations into one overall transformation. When applying these changes, it is important to follow the correct order of operations, as the sequence in which transformations are applied can affect the final outcome of the graph.

Transcript

00:01 The question gives us the equation, y equals cosine of x, and gives us many transformations with it.

00:08 So we have to stretch it by a factor of three, move it pi units to the right, move two units down, and reflect it over the x -axis.

00:18 And then the question asks us to find the equation of the curve after all of the transformations.

00:25 So first to do that, we need to figure out what these transformations would do to the equation.

00:30 So stretching by factor of three, it means that we take the amplitude and triple it or multiply it by three.

00:42 So our amplitude is one right now since there is no coefficient from the cosine.

00:47 So we triple it.

00:48 So our amplitude will be three.

00:50 Next, we move pi units to the right, which means that our phase shift, or we add pi to our phase shift.

01:00 And right now our phase shift is zero.

01:01 So our phase shift after the transformation is going to be pi.

01:07 Next, we have to move two units down.

01:10 And to do that, we make our vertical shift negative 2.

01:20 And finally, we reflected over the x -axis, which would actually be slightly tricky.

01:27 And for me to explain that, i need to draw a quick graph over here.

01:33 So right now, our function is going to be down here.

01:37 And look something like this.

01:41 When we reflect over the x -axis, we are not going to be just flipping the amplitude.

01:47 So it looks like this.

01:50 Instead, we're actually going to be flipping the vertical shift as well.

01:55 So this entire part, this entire equation is going to be flipped over...

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