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Algebra II : Transformations

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Example Question #1 : Transformations

$f(x)=x^{2}$

$g(x)=4x^{2}-3$

How is the graph of g(x) different from the graph of f(x) ?

Possible Answers:

g(x) is wider than f(x) and is shifted down 3 units

g(x) is wider than f(x) and is shifted to the right 3 units

g(x) is narrower than f(x) and is shifted down 3 units

g(x) is narrower than f(x) and is shifted up 3 units

g(x) is narrower than f(x) and is shifted to the left 3 units

Correct answer:

g(x) is narrower than f(x) and is shifted down 3 units

Explanation:

Almost all transformed functions can be written like this:

g(x)=a[f(b(x-c))]+d

where f(x) is the parent function. In this case, our parent function is $f(x)=x^{2}$ , so we can write g(x) this way:

$g(x)=a[b(x-c)]^{2}+d$

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.

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Example Question #2 : Transformations

Which of the following transformations represents a parabola shifted to the right by 4 and halved in width?

Possible Answers:

f(x) = 2(x-4)^2

f(x)=2x^2+4

f(x) = 2x^2 - 4

f(x) = 0.5(x-4)^2

f(x) = 2(x+4)^2

Correct answer:

f(x) = 2(x-4)^2

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x)=(x-4)^2

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.

f(x) = 2(x-4)^2

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Example Question #1 : Transformations

Which of the following represents a standard parabola shifted up by 2 units?

Possible Answers:

f(x)=(x-2)^2 - 4

f(x)=(x-2)^2 + 2

f(x)=(x+2)^2

f(x)=x^2 + 2

f(x)=(x-2)^2

Correct answer:

f(x)=x^2 + 2

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x)=x^2+2

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Example Question #4 : Transformations

Which of the following transformation flips a parabola vertically, doubles its width, and shifts it up by 3?

Possible Answers:

f(x)=-(0.5x^2 +3)

f(x)=-0.5x^2 +3

f(x)=-2(x^2 +3)

f(x)=-2x^2 +3

f(x)=-0.5x^2 +1.5

Correct answer:

f(x)=-0.5x^2 +3

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x)=-x^2

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.

$f(x)=-(\frac{1}{2})x^2=-0.5x^2$

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.

f(x)=-0.5x^2 +3

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Example Question #5 : Transformations

Which of the following shifts a parabola six units to the right and five downward?

Possible Answers:

f(x)=(x+6)^2-5

$f(x)=(x-\sqrt{5})^2+6$

$f(x)=(x-\sqrt{6})^2-5$

f(x)=(x-6)^2-5

f(x)=(x-5)^2-6

Correct answer:

f(x)=(x-6)^2-5

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x)=x^2-5

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.

f(x)=(x-6)^2 - 5

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Example Question #6 : Transformations

Which of the following transformations represents a parabola that has been flipped vertically, shifted to the right 12, and shifted downward 4?

Possible Answers:

f(x)=-(x+12)^2+4

f(x)=-(x-12)^2-4

f(x)=-[(x+12)^2-4]

f(x)=-[(x+12)^2+4]

f(x)=-(x+12)^2-4

Correct answer:

f(x)=-(x-12)^2-4

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x) = (x-12)^2

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.

f(x) = -(x-12)^2

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.

f(x) = -(x-12)^2 - 4

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Example Question #7 : Transformations

Which of the following transformations represents a parabola that has been shifted 4 units to the left, 5 units down, and quadrupled in width?

Possible Answers:

f(x) = 4(x-4)^2-5

f(x) = 4(x+4)^2-5

f(x) = 0.25(x-4)^2-5

f(x) = 0.25(x+4)^2-5

f(x) = 0.25(x-4)^2+5

Correct answer:

f(x) = 0.25(x+4)^2-5

Explanation:

Begin with the standard equation for a parabola: f(x)=x^2 .

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.

f(x) = (x+4)^2

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.

f(x) = 0.25(x+4)^2 - 5

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Example Question #1 : Transformations

If the function y=1-x is shifted left 2 units, and up 3 units, what is the new equation?

Possible Answers:

y=x-2

y=4-x

y=-1-x

y=3-x

y=2-x

Correct answer:

y=2-x

Explanation:

Shifting y=1-x left 2 units will change the y-intercept from (0,1) to (0.-1) .

The new equation after shifting left 2 units is:

y=1-(x+2)=-1-x

Shifting up 3 units will add 3 to the y-intercept of the new equation.

The answer is: y=2-x

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Example Question #9 : Transformations

If f(x)=x^2 , what is -f(x) ?

Possible Answers:

It is the $x^{-2}$ function.

It is the x^2 parabola shifted to the right by 1.

It is the x^2 parabola reflected across the x-axis.

It is the same as f(x) .

It is the x^2 parabola reflected across the y-axis.

Correct answer:

It is the x^2 parabola reflected across the x-axis.

Explanation:

It helps to evaluate the expression algebraically.

-f(x)=-x^2 . 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 x^2

X 2

and this is the graph of -x^2

x 2

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Example Question #10 : Transformations

If f(x)=e^x , what is f(-x) ?

Possible Answers:

It is the e^x graph shifted 1 to the right.

It is the e^x graph reflected across the y-axis.

It is the e^x graph reflected across the x-axis.

It is the e^x graph rotated about the origin.

It is the -e^x graph.

Correct answer:

It is the e^x graph reflected across the y-axis.

Explanation:

Algebraically, $f(-x)=e^{-x}$ .

This is a reflection across the y axis.

This is the graph of e^x

E x

And this is the graph of $e^{-x}$

E x

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Algebra II : Transformations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Transformations

Example Question #2 : Transformations

Example Question #1 : Transformations

Example Question #4 : Transformations

Example Question #5 : Transformations

Example Question #6 : Transformations

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