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Algebra II : Algebra II

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Example Questions

Example Question #171 : Introduction To Functions

Which of the following is the equation of a vertical asymptote of the graph of the function $f(x) = \frac{x+7}{x^{2}- 16}$ ?

(a) x= -7

(b) x= -4

Possible Answers:

All three of (a), (b), and (c)

(b) only

(b) and (c) only

(a) only

Correct answer:

(b) and (c) only

Explanation:

The vertical asymptote(s) of the graph of a rational function such as f(x) can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

$x^{2} - 16 = 0$

Add 16 to both sides:

$x^{2} - 16 + 16 = 0+ 16$

$x^{2} =16$

Take the positive and negative square roots:

$x = -\sqrt{16 } = -4$

$x = \sqrt{16 } = 4$

The graph of f(x) has two vertical asymptotes, the graphs of the lines x = -4 and x= 4 .

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Example Question #174 : Functions And Graphs

If g(x)=5x+6x^2+3 and f(x)=x-4 what is g(f(x)) ?

Possible Answers:

6x^2+5x+79

6x^2-53x+119

6x^2-43x+79

6x^2+5x-1

Correct answer:

6x^2-43x+79

Explanation:

g(f(x)) is a composite function which means that the inside function is plugged into the outside function. So in this case, f(x) is plugged into g(x) . In other words, replace the f(x) expression each time there is an in the g(x) expression.

In this case x-4 would be plugged into each in the g(x) expression. See below:

5(x-4)+6(x-4)^2+3

This is then simplified to:

5x-20+6x^2-48x+96+3

And then further simplified to:

6x^2-43x+79

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

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

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 up 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) = 0.5(x-4)^2

f(x) = 2x^2 - 4

f(x)=2x^2+4

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 #2 : Transformations

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

Possible Answers:

f(x)=x^2 + 2

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

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

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

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

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

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

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 #1 : Transformations

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

Possible Answers:

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

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

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

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 #181 : Functions And Graphs

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 #182 : Functions And Graphs

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

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

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 #2 : Transformations

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

Possible Answers:

y=4-x

y=-1-x

y=3-x

y=2-x

y=x-2

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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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #171 : Introduction To Functions

Example Question #174 : Functions And Graphs

Example Question #1 : Transformations

Example Question #2 : Transformations

Example Question #2 : Transformations

Example Question #4 : Transformations

Example Question #1 : Transformations

Example Question #181 : Functions And Graphs

Example Question #182 : Functions And Graphs

Example Question #2 : Transformations

All Algebra II Resources

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