Transformations - Algebra II
Card 0 of 80
How is the graph of 
different from the graph of 
?
How is the graph of
Almost all transformed functions can be written like this:
where 
is the parent function. In this case, our parent function is 
, so we can write 
this way:
Luckily, for this problem, we only have to worry about 
and 
.
represents the vertical stretch factor of the graph.
- If
is less than 1, the graph has been vertically compressed by a factor of 
. It's almost as if someone squished the graph while their hands were on the top and bottom. This would make a parabola, for example, look wider.
- If
is greater than 1, the graph has been vertically stretched by a factor of 
. It's almost as if someone pulled on the graph while their hands were grasping the top and bottom. This would make a parabola, for example, look narrower.
represents the vertical translation of the graph.
- If
is positive, the graph has been shifted up 
units.
- If
is negative, the graph has been shifted down 
units.
For this problem, 
is 4 and 
is -3, causing vertical stretch by a factor of 4 and a vertical translation down 3 units.
Almost all transformed functions can be written like this:
where
Luckily, for this problem, we only have to worry about
- If
- If
- If
- If
For this problem,
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Which of the following represents a standard parabola shifted up by 2 units?
Which of the following represents a standard parabola shifted up by 2 units?
Begin with the standard equation for a parabola: 
.
Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 2 units, add 2.
Begin with the standard equation for a parabola:
Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 2 units, add 2.
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Which of the following transformation flips a parabola vertically, doubles its width, and shifts it up by 3?
Which of the following transformation flips a parabola vertically, doubles its width, and shifts it up by 3?
Begin with the standard equation for a parabola: 
.
Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the 
. If the coefficient is negative, then the parabola opens downward.
The width of the parabola is determined by the magnitude of the coefficient in front of 
. To make a parabola wider, use a fractional coefficient. Doubling the width indicates a coefficient of one-half.
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 3 units, add 3.
Begin with the standard equation for a parabola:
Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the
The width of the parabola is determined by the magnitude of the coefficient in front of
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift upward 3 units, add 3.
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Which of the following shifts a parabola six units to the right and five downward?
Which of the following shifts a parabola six units to the right and five downward?
Begin with the standard equation for a parabola: .
Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the 
term. To shift 6 units to the right, subtract 6 within the parenthesis.
Begin with the standard equation for a parabola:
Vertical shifts to this standard equation are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the
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Which of the following transformations represents a parabola shifted to the right by 4 and halved in width?
Which of the following transformations represents a parabola shifted to the right by 4 and halved in width?
Begin with the standard equation for a parabola: .
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the term. To shift 4 units to the right, subtract 4 within the parenthesis.
The width of the parabola is determined by the magnitude of the coefficient in front of . To make a parabola narrower, use a whole number coefficient. Halving the width indicates a coefficient of two.
Begin with the standard equation for a parabola:
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the
The width of the parabola is determined by the magnitude of the coefficient in front of
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Which of the following transformations represents a parabola that has been flipped vertically, shifted to the right 12, and shifted downward 4?
Which of the following transformations represents a parabola that has been flipped vertically, shifted to the right 12, and shifted downward 4?
Begin with the standard equation for a parabola: .
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the 
term. To shift 12 units to the right, subtract 12 within the parenthesis.
Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the 
. If the coefficient is negative, then the parabola opens downward.
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 4 units, subtract 4.
Begin with the standard equation for a parabola:
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the
Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 4 units, subtract 4.
Compare your answer with the correct one above
Which of the following transformations represents a parabola that has been shifted 4 units to the left, 5 units down, and quadrupled in width?
Which of the following transformations represents a parabola that has been shifted 4 units to the left, 5 units down, and quadrupled in width?
Begin with the standard equation for a parabola: .
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the 
term. To shift 4 units to the left, add 4 within the parenthesis.
The width of the parabola is determined by the magnitude of the coefficient in front of 
. To make a parabola wider, use a fractional coefficient. Doubling the width indicates a coefficient of one-fourth.
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.
Begin with the standard equation for a parabola:
Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the
The width of the parabola is determined by the magnitude of the coefficient in front of
Vertical shifts are represented by additions (upward shifts) or subtractions (downward shifts) added to the end of the equation. To shift downward 5 units, subtract 5.
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If the function 
is shifted left 2 units, and up 3 units, what is the new equation?
If the function
Shifting 
left 2 units will change the y-intercept from 
to 
.
The new equation after shifting left 2 units is:
Shifting up 3 units will add 3 to the y-intercept of the new equation.
The answer is: 
Shifting
The new equation after shifting left 2 units is:
Shifting up 3 units will add 3 to the y-intercept of the new equation.
The answer is:
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Transformations
Where will the point 
be located after the following transformations?
- Reflection about the x-axis
- Translation up 3
- Translation right 4
Transformations
Where will the point
- Reflection about the x-axis
- Translation up 3
- Translation right 4
Where will the point 
be located after the following transformations?
- Reflection about the x-axis results in multiplying the y value by negative one thus
.
- Translation up 3, means to add three to the y values which results in
.
- Translation right 4, means to add four to the x value which will result in
.
Where will the point
- Reflection about the x-axis results in multiplying the y value by negative one thus
- Translation up 3, means to add three to the y values which results in
- Translation right 4, means to add four to the x value which will result in
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If 
, what is 
?
If
It helps to evaluate the expression algebraically.
. Any time you multiply a function by a -1, you reflect it over the x axis. It helps to graph for verification.
This is the graph of 
and this is the graph of 
It helps to evaluate the expression algebraically.
This is the graph of
and this is the graph of
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If 
, what is 
?
If
Algebraically, 
.
This is a reflection across the y axis.
This is the graph of 
And this is the graph of 
Algebraically,
This is a reflection across the y axis.
This is the graph of
And this is the graph of
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Find the equation of the linear function 
obtained by shifting the following linear function 
along the x-axis 3 units to the left. State the y-intercept of 
.
Find the equation of the linear function
The transformation for a left shift along the x-axis for 
requires we add 
to the argument of the function 
.
The y-intercept of the linear function 
is 
.
The transformation for a left shift along the x-axis for
The y-intercept of the linear function
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If the function 
is linear and the function 
is quadratic, then the function 
is?
If the function
The linear function 
will have the form,
Where 
is the y-intercept and 
is the slope; both are constant.
The quadratic function 
will have the form,
We are given that the function 
is defined,
we obtain another function that is also a quadratic function since 
and 
are constants. Therefore, 
is quadratic.
The linear function
Where
The quadratic function
We are given that the function
we obtain another function that is also a quadratic function since
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Reflect 
across the x-axis, then reflect across 
, and then shift this line up five units. What is the new equation?
Reflect
Reflect 
across the x-axis will turn the equation to:
If we then reflect 
across 
, the equation will become:
Shifting this line up five units means that we will add five to this equation.
The equation after all the transformations is: 
The answer is: 
Reflect
If we then reflect
Shifting this line up five units means that we will add five to this equation.
The equation after all the transformations is:
The answer is:
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Shift 
down three units. What is the new equation?
Shift
The equation is currently in standard form. Rewrite the current equation in slope-intercept form. Subtract 
from both sides.
Since the graph is shifted downward three units, all we need to change is decrease the y-intercept by three. Subtract the right side by three.
The answer is: 
The equation is currently in standard form. Rewrite the current equation in slope-intercept form. Subtract
Since the graph is shifted downward three units, all we need to change is decrease the y-intercept by three. Subtract the right side by three.
The answer is:
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Reflect the line 
across the line 
, and then reflect again across the line 
. What is the new equation of this horizontal line?
Reflect the line
The distance between 
and 
is three units. If the line 
is reflected across 
, this means that the new line will also be three units away from 
.
The equation of the line after this reflection is: 
If this line is reflected again across the line 
, both lines are six units apart, and the reflection would mean that the line below line 
is also six units apart.
Subtract six from line 
.
The equation of the line after the transformations is: 
The answer is: 
The distance between
The equation of the line after this reflection is:
If this line is reflected again across the line
Subtract six from line
The equation of the line after the transformations is:
The answer is:
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If the function 
is shifted down two units and left four units, what is the new y-intercept?
If the function
If the graph was shifted two units down, only the y-intercept will change, and will decrease by two.
The new equation is: 
If the graph was shifted left four units, the root will shift four units to the left, and the 
will need to be replaced with 
.
The new y-intercept will be 
.
If the graph was shifted two units down, only the y-intercept will change, and will decrease by two.
The new equation is:
If the graph was shifted left four units, the root will shift four units to the left, and the
The new y-intercept will be
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Reflect the line 
across 
, and shift the line down three units. What is the new equation?
Reflect the line
The equations with an existing 
variable is incorrect because they either represent lines with slopes or vertical lines.
After the line 
is reflected across 
, the line becomes 
.
Shifting this line down three units mean that the line will have a vertical translation down three.
Subtract the equation 
by three.
The result is: 
The equations with an existing
After the line
Shifting this line down three units mean that the line will have a vertical translation down three.
Subtract the equation
The result is:
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Give the equation of the vertical asymptote of the graph of the equation 
.
Give the equation of the vertical asymptote of the graph of the equation
Define 
. As an exponential function, this has a graph that has no vertical asymptote, as 
is defined for all real values of 
. In terms of 
:
,
The graph of 
is a transformation of that of 
- a horizontal shift ( 
), a vertical stretch ( 
), and a vertical shift ( 
) of the graph of 
; none of these transformations changes the status of the function as one whose graph has no vertical asymptote.
Define
The graph of
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Give the equation of the horizontal asymptote of the graph of the equation
Give the equation of the horizontal asymptote of the graph of the equation
Define 
in terms of ,
It can be restated as the following:
The graph of 
has as its horizontal asymptote the line of the equation 
. The graph of is a transformation of that of —a right shift of 2 units 
, a vertical stretch 
, and an upward shift of 5 units 
. The right shift and the vertical stretch do not affect the position of the horizontal asymptote, but the upward shift moves the asymptote to the line of the equation 
. This is the correct response.
Define
It can be restated as the following:
The graph of
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