# Algebra II : Transformations

## Example Questions

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### Example Question #1 : Transformations

How is the graph of  different from the graph of ?

is narrower than and is shifted up 3 units

is wider than and is shifted down 3 units

is wider than and is shifted to the right 3 units

is narrower than and is shifted down 3 units

is narrower than and is shifted to the left 3 units

is narrower than and is shifted down 3 units

Explanation:

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.

### Example Question #2 : Transformations

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

Explanation:

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.

### Example Question #3 : Transformations

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

Explanation:

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.

### Example Question #4 : Transformations

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

Explanation:

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.

### Example Question #5 : Transformations

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

Explanation:

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.

### 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?

Explanation:

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.

### 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?

Explanation:

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.

### Example Question #8 : Transformations

If the function  is shifted left 2 units, and up 3 units, what is the new equation?

Explanation:

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.

### Example Question #9 : Transformations

If , what is ?

It is the  parabola shifted to the right by 1.

It is the  parabola reflected across the x-axis.

It is the same as .

It is the  parabola reflected across the y-axis.

It is the  function.

It is the  parabola reflected across the x-axis.

Explanation:

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

### Example Question #10 : Transformations

If , what is ?

It is the  graph rotated about the origin.

It is the  graph reflected across the y-axis.

It is the  graph reflected across the x-axis.

It is the  graph.

It is the  graph shifted 1 to the right.

It is the  graph reflected across the y-axis.

Explanation:

Algebraically, .

This is a reflection across the y axis.

This is the graph of

And this is the graph of

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