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Algebra II : Transformations

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Example Questions

Example Question #1 : Transformations

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

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Almost all transformed functions can be written like this:

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

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

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

Luckily, for this problem, we only have to worry about $\displaystyle a$ and $\displaystyle d$.

$\displaystyle a$ represents the vertical stretch factor of the graph.

If $\displaystyle |a|$ is less than 1, the graph has been vertically compressed by a factor of $\displaystyle |a|$. 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 $\displaystyle |a|$ is greater than 1, the graph has been vertically stretched by a factor of $\displaystyle |a|$. 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.

$\displaystyle d$ represents the vertical translation of the graph.

If $\displaystyle d$ is positive, the graph has been shifted up $\displaystyle d$ units.
If $\displaystyle d$ is negative, the graph has been shifted down $\displaystyle d$ units.

For this problem, $\displaystyle a$ is 4 and $\displaystyle d$ 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:

$\displaystyle f(x) = 0.5(x-4)^2$

$\displaystyle f(x) = 2x^2 - 4$

$\displaystyle f(x)=2x^2+4$

$\displaystyle f(x) = 2(x-4)^2$

$\displaystyle f(x) = 2(x+4)^2$

Correct answer:

$\displaystyle f(x) = 2(x-4)^2$

Explanation:

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

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 4 units to the right, subtract 4 within the parenthesis.

$\displaystyle f(x)=(x-4)^2$

The width of the parabola is determined by the magnitude of the coefficient in front of $\displaystyle x$. To make a parabola narrower, use a whole number coefficient. Halving the width indicates a coefficient of two.

$\displaystyle f(x) = 2(x-4)^2$

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Example Question #3 : Transformations

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

Possible Answers:

$\displaystyle f(x)=x^2 + 2$

$\displaystyle f(x)=(x-2)^2 - 4$

$\displaystyle f(x)=(x-2)^2$

$\displaystyle f(x)=(x-2)^2 + 2$

$\displaystyle f(x)=(x+2)^2$

Correct answer:

$\displaystyle f(x)=x^2 + 2$

Explanation:

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

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.

$\displaystyle f(x)=x^2+2$

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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:

$\displaystyle f(x)=-(0.5x^2 +3)$

$\displaystyle f(x)=-2(x^2 +3)$

$\displaystyle f(x)=-2x^2 +3$

$\displaystyle f(x)=-0.5x^2 +1.5$

$\displaystyle f(x)=-0.5x^2 +3$

Correct answer:

$\displaystyle f(x)=-0.5x^2 +3$

Explanation:

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

Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the $\displaystyle x$. If the coefficient is negative, then the parabola opens downward.

$\displaystyle f(x)=-x^2$

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

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.

$\displaystyle f(x)=-0.5x^2 +3$

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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:

$\displaystyle f(x)=(x-6)^2-5$

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

$\displaystyle f(x)=(x+6)^2-5$

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

$\displaystyle f(x)=(x-5)^2-6$

Correct answer:

$\displaystyle f(x)=(x-6)^2-5$

Explanation:

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

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.

$\displaystyle f(x)=x^2-5$

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 6 units to the right, subtract 6 within the parenthesis.

$\displaystyle f(x)=(x-6)^2 - 5$

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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:

$\displaystyle f(x)=-(x+12)^2-4$

$\displaystyle f(x)=-(x-12)^2-4$

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

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

$\displaystyle f(x)=-(x+12)^2+4$

Correct answer:

$\displaystyle f(x)=-(x-12)^2-4$

Explanation:

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

Horizontal shifts are represented by additions (leftward shifts) or subtractions (rightward shifts) within the parenthesis of the $\displaystyle x$ term. To shift 12 units to the right, subtract 12 within the parenthesis.

$\displaystyle f(x) = (x-12)^2$

Inverting, or flipping, a parabola refers to the sign in front of the coefficient of the $\displaystyle x$. If the coefficient is negative, then the parabola opens downward.

$\displaystyle f(x) = -(x-12)^2$

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.

$\displaystyle f(x) = -(x-12)^2 - 4$

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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:

$\displaystyle f(x) = 4(x-4)^2-5$

$\displaystyle f(x) = 0.25(x-4)^2-5$

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

$\displaystyle f(x) = 4(x+4)^2-5$

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

Correct answer:

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

Explanation:

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

$\displaystyle f(x) = (x+4)^2$

The width of the parabola is determined by the magnitude of the coefficient in front of $\displaystyle x$. To make a parabola wider, use a fractional coefficient. Doubling the width indicates a coefficient of one-fourth. $f(x) = \frac{1}{4}(x+4)^2=0.25(x+4)^2$ $\displaystyle f(x) = \frac{1}{4}(x+4)^2=0.25(x+4)^2$

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.

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

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Example Question #8 : Transformations

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

Possible Answers:

y=4-x $\displaystyle y=4-x$

y=3-x $\displaystyle y=3-x$

y=2-x $\displaystyle y=2-x$

y=x-2 $\displaystyle y=x-2$

y=-1-x $\displaystyle y=-1-x$

Correct answer:

y=2-x $\displaystyle y=2-x$

Explanation:

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

The new equation after shifting left 2 units is:

$\displaystyle y=1-(x+2)=-1-x$

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

The answer is: y=2-x $\displaystyle y=2-x$

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Example Question #9 : Transformations

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

Possible Answers:

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

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

It is the same as f(x) $\displaystyle f(x)$.

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

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

Correct answer:

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

Explanation:

It helps to evaluate the expression algebraically.

$\displaystyle -f(x)=-x^2$. 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 $\displaystyle x^2$

X 2

and this is the graph of -x^2 $\displaystyle -x^2$

x 2

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Example Question #10 : Transformations

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

Possible Answers:

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

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

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

It is the e^x $\displaystyle e^x$ graph rotated about the origin.

It is the -e^x $\displaystyle -e^x$ graph.

Correct answer:

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

Explanation:

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

This is a reflection across the y axis.

This is the graph of e^x $\displaystyle e^x$

E x

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

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

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