Algebra II - Transformations

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

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

How is the graph of different from the graph of ?

is narrower than and is shifted down 3 units

is wider than and is shifted down 3 units

is narrower than and is shifted up 3 units

is wider than and is shifted to the right 3 units

is narrower than and is shifted to the left 3 units

Explanation

Almost all transformed functions can be written like this:

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

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.

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.

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

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.

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

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

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

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.

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.

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.

If , what is ?

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 rotated about the origin.

Explanation

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

This is a reflection across the y axis.

This is the graph of

E x

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

E x

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.

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.

Give the equation of the horizontal asymptote of the graph of the equation $f(x) = 4e^{x-2}+ 5$

The graph of has no horizontal asymptote.

Explanation

Define $g(x) = e ^{x}$ 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.