All HiSET: Math Resources
Example Questions
Example Question #1 : Reflections
What is the result of reflecting the point over the y-axis in the coordinate plane?
Reflecting a point
over the y-axis geometrically is the same as negating the x-coordinate of the ordered pair to obtain
.
Thus, since our initial point was
and we want to reflect it over the y-axis, we obtain the reflection by negating the first term of the ordered pair to get
.
Example Question #1 : Understand Transformations In The Plane
How many lines of symmetry does the above figure have?
Two
Infinitely many
None
Four
One
One
A line of symmetry of a figure is about which the reflection of the figure is the figure itself. The diagram below shows the only line of symmetry of the figure.
The correct response is one.
Example Question #1 : Reflections
On the coordinate plane, the point is reflected about the
-axis. The image is denoted
. Give the distance
.
The reflection of the point at about the
-axis is the point at
; therefore, the image of
is
. Since these two points have the same
-coordinate, the distance between them is the absolute value of the difference between their
-coordinates:
Example Question #41 : Measurement And Geometry
Consider regular Hexagon ; let
and
be the midpoints of
and
. Reflect the hexagon about
, then again about
. Which of the following clockwise rotations about the center would result in each point being its own image under this series of transformations?
Refer to the figure below, which shows the reflection of the given hexagon about ; we will call
the image of
, call
the image of
, and so forth.
Now, refer to the figure below, which shows the reflection of the image about ; we will call
the image of
, call
the image of
, and so forth.
Note that the vertices coincide with those of the original hexagon, and that the images of the points are in the same clockwise order as the original points. Since coincides with
,
coincides with
, and so forth, a clockwise rotation of five-sixth of a complete turn - that is,
,
is required to make each point its own image under the three transformations.
Example Question #1 : Understand Transformations In The Plane
Consider regular Hexagon ; let
and
be the midpoints of
and
. Reflect the hexagon about
, then again about
. With which of the following points does the image of
under these reflections coincide?
Refer to the figure below, which shows the reflection of about
; we will call this image
.
Note that coincides with
. Now, refer to the figure below, which shows the reflection of
about
; we will call this image - the final image -
Note that coincides with
, making this the correct response.
Example Question #1 : Translations
The graph on the left shows an object in the Cartesian plane. A transformation is performed on it, resulting in the graph on the right.
Which of the following transformations best fits the graphs?
Reflection in the y-axis
Rotation about the origin
Reflection in the x-axis
Dilation
Translation
Translation
A dilation is the stretching or shrinking of a figure.
A rotation is the turning of a a figure about a point.
A reflection is the flipping of a figure across a line.
A translation is is the sliding of a figure in a direction.
With a translation, the image is not only congruent to its original size and shape, but its orientation remains the same. A translation fits this figure best because the shape seems to move upward and rightward without changing size, shape, or orientation.
Example Question #1 : Understand Transformations In The Plane
Translate the graph of the equation
left four units and down six units. Give the equation of the image.
If the graph of an equation is translated to the right units, and upward
units, the equation of the image can be found by replacing
with
and
with
in the equation of the original graph.
Since we are moving the graph of the equation
left four units and down six units, we set and
; we can replace
with
, or
, and
with
, or
. The equation of the image can be written as
Simplify by distributing:
Collect like terms:
Add 26 to both sides:
,
the correct choice.
Example Question #1 : Understand Transformations In The Plane
Translate the graph of the equation
right two units and up five units. Give the equation of the image.
None of the other choices gives the correct response.
If the graph of an equation is translated to the right units, and upward
units, the equation of the image can be found by replacing
with
and
with
in the equation of the original graph.
Since we are moving the graph of the equation
right two units and up five units, we set and
; we can therefore replace
with
and
with
. The equation of the image can be written as
This can be rewritten by applying the binomial square pattern as follows:
Collect like terms; the equation becomes
Subtract 100 from both sides:
Example Question #1 : Translations
Translate the graph of the equation
right four units and down two units. Give the equation of the image.
If the graph of an equation is translated to the right units, and upward
units, the equation of the image can be found by replacing
with
and
with
in the equation of the original graph.
Since we are moving the graph of the equation
right four units and down two units, we set and
; we can therefore replace
with
, and
with
, or
. The equation of the image can be written as
The expression at right can be simplified. First, use the distributive property on the middle expression:
Now, simplify the first expression by using the binomial square pattern:
Collect like terms on the right:
Subtract 2 from both sides:
,
the equation of the image.
Example Question #1 : Understand Transformations In The Plane
Translate the graph of the equation
left three units and down five units. Give the equation of the image.
None of the other choices gives the correct response.
If the graph of an equation is translated to the right units, and upward
units, the equation of the image can be found by replacing
with
and
with
in the equation of the original graph.
Since we are moving the graph of the equation
left three units and down five units, we set and
; we can therefore replace
with
, or
, and
with
, or
. The equation of the image can be written as
We can simplify the expression on the right by distributing:
Collect like terms:
Subtract 5 from both sides:
,
the correct equation of the image.
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