HOW DOYOU SEE IT? Use the plot of the point (x0, y0) in the figure. Match the transformation of

HOW DOYOU SEE IT? Use the plot of the point (x0, y0) in the...

HOW DOYOU SEE IT? Use the plot of the point $\left(x_{0}, y_{0}\right)$ in the figure. Match the transformation of the point with the correct plot. Explain your reasoning. IThe plots are labeled (i), (ii), (iii), and (iv).] $$ \begin{array}{ll}{\text { (a) }\left(x_{0}, y_{0}\right)} & {\text { (b) }\left(-2 x_{0}, y_{0}\right)} \\ {\text { (c) }\left(x_{0}, \frac{1}{2} y_{0}\right)} & {\text { (d) }\left(-x_{0},-y_{0}\right)}\end{array} $$

HOW DOYOU SEE IT? Use the plot of
the point $\left(x_{0}, y_{0}\right)$ in the figure. Match the
transformation of the point with the correct
plot. Explain your reasoning. IThe plots are
labeled (i), (ii), (iii), and (iv).]
$$
\begin{array}{ll}{\text { (a) }\left(x_{0}, y_{0}\right)} & {\text { (b) }\left(-2 x_{0}, y_{0}\right)} \\ {\text { (c) }\left(x_{0}, \frac{1}{2} y_{0}\right)} & {\text { (d) }\left(-x_{0},-y_{0}\right)}\end{array}
$$

Key Concepts

Coordinate Transformations

Coordinate transformations are operations that map points from one position in the Cartesian plane to another by applying specific rules to the x and y coordinates. This concept encompasses a variety of actions such as translations, rotations, reflections, and dilations, which change the location, orientation, or size of a figure while preserving some of its essential properties.

Reflection Transformations

Reflection transformations involve 'flipping' a point or figure over a specific line or point, such as the x-axis, y-axis, or the origin. When a reflection is applied, the coordinates are modified by changing the signs of one or both components, which results in a mirror image of the original object across the axis or point of reflection.

Dilation (Scaling) Transformations

Dilation or scaling transformations change the size of an object by multiplying the coordinates by a constant factor. This operation can affect the x-coordinate, the y-coordinate, or both, and can either enlarge or shrink the object. A negative scaling factor can also introduce a reflection component along with the size change.

Composite Transformations

Composite transformations involve applying more than one type of transformation in sequence. When multiple operations such as reflection and scaling are combined, the resulting transformation can be seen as the outcome of applying one transformation followed by another, and it is important to understand how each individual transformation contributes to the final coordinates of the point.

Transcript

00:04 We are given a transformation in each part, and we are asked to match the transformation of a given point with the correct plot, and to explain.

00:22 So in exercise 58, we are given the figure contains the plot of the point x0 y0.

00:36 Now, in part a, the transformation of the point is x0y0 gets mapped to x0y0 again.

00:59 That is the figure should stay the same.

01:15 So the correct plot will be the same as the original figure.

01:28 Therefore, plot 4 is the correct, since it's the same as the original plot.

01:40 Next in part b, we're told that x0 y0 is transformed into negative 2x0 y0.

01:57 So two things.

02:03 We see that y0 remains the same.

02:12 This means that we stay on the line y equals y0.

02:22 However, x 0, the negative 2 means that x 0 is flipped across the origin 0 and stretched by a factor of 2.

03:13 So we have that x0 y0 is stretched by a factor of 2.

03:27 In the x direction and reflected across the y axis therefore the correct plot is plot three in part c we're told that x0 y0 was transformed into x 0 1 1 half y 0 we see that this transformation tells us that x0 y0 is compressed vertically by a factor of 2.

04:39 You could also say stretched by a factor of 1 half.

05:23 I'm thinking a mistake was made in this problem, because it looks like this should actually be plot 4...

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