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  1. AP Pre Calculus
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What are the differences between vertical and horizontal translations?

Vertical: Shifts up/down, affects y-values. Horizontal: Shifts left/right, affects x-values.

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What are the differences between vertical and horizontal translations?

Vertical: Shifts up/down, affects y-values. Horizontal: Shifts left/right, affects x-values.

What are the differences between vertical and horizontal dilations?

Vertical: Stretches/shrinks vertically, affects y-values. Horizontal: Stretches/shrinks horizontally, affects x-values.

Compare reflection over the x-axis vs. y-axis.

X-axis: Flips over x-axis, negates y-values. Y-axis: Flips over y-axis, negates x-values.

Contrast the effects of f(x)+kf(x) + kf(x)+k and f(x+k)f(x + k)f(x+k).

f(x)+kf(x) + kf(x)+k: Vertical shift by k. f(x+k)f(x + k)f(x+k): Horizontal shift by -k.

Compare the effects of af(x)af(x)af(x) and f(ax)f(ax)f(ax).

af(x)af(x)af(x): Vertical dilation by a. f(ax)f(ax)f(ax): Horizontal dilation by 1/a.

What is the difference between a vertical stretch and a vertical shift?

Vertical stretch: Changes the shape of the graph by multiplying y-values. Vertical shift: Moves the graph up or down without changing its shape.

How do horizontal stretches and compressions differ?

Horizontal stretch: Expands the graph horizontally. Horizontal compression: Shrinks the graph horizontally.

Compare the effects of a positive vs. negative 'a' in g(x)=af(x)g(x) = af(x)g(x)=af(x).

Positive 'a': Vertical stretch or compression. Negative 'a': Vertical stretch or compression AND reflection over the x-axis.

Contrast the impact of 'h' and 'k' in the general transformation equation.

'h': Horizontal shift (left/right). 'k': Vertical shift (up/down).

How does changing 'b' in f(bx)f(bx)f(bx) affect the graph differently than changing 'a' in af(x)af(x)af(x)?

Changing 'b' affects the horizontal aspect (stretch/compression/reflection over y-axis), while changing 'a' affects the vertical aspect (stretch/compression/reflection over x-axis).

Explain the effect of k in f(x)+kf(x) + kf(x)+k.

Shifts the graph vertically. k>0k > 0k>0 moves the graph up, k<0k < 0k<0 moves it down.

Explain the effect of h in f(x+h)f(x + h)f(x+h).

Shifts the graph horizontally. h>0h > 0h>0 moves the graph left, h<0h < 0h<0 moves it right.

Explain the effect of 'a' in af(x)af(x)af(x).

Scales the graph vertically. ∣a∣>1|a| > 1∣a∣>1 stretches, 0 < |a| < 1 shrinks. a<0a < 0a<0 reflects over x-axis.

Explain the effect of 'b' in f(bx)f(bx)f(bx).

Scales the graph horizontally. ∣b∣>1|b| > 1∣b∣>1 shrinks, 0 < |b| < 1 stretches. b<0b < 0b<0 reflects over y-axis.

Why does horizontal translation appear 'opposite'?

Because f(x+h)f(x+h)f(x+h) evaluates the function at a shifted x-value, requiring a shift in the opposite direction to achieve the same output.

How do transformations affect the domain and range?

Translations shift the domain/range, dilations compress/expand them, and reflections can change the direction of the range.

What is the order of transformations?

Horizontal transformations (shifts and stretches) before vertical transformations.

Explain how a vertical stretch affects the range of a function.

A vertical stretch multiplies the range values by the stretch factor, expanding or compressing the range.

Describe the impact of a negative 'a' value in g(x)=af(x)g(x) = af(x)g(x)=af(x).

It reflects the graph of f(x)f(x)f(x) over the x-axis, changing the sign of the y-values.

What happens to the x-intercepts after a vertical stretch?

The x-intercepts remain unchanged because the y-value at these points is zero, and multiplying zero by any factor still results in zero.

Vertical translation by k units:

g(x)=f(x)+kg(x) = f(x) + kg(x)=f(x)+k

Horizontal translation by h units:

g(x)=f(x+h)g(x) = f(x + h)g(x)=f(x+h)

Vertical dilation by a factor of a:

g(x)=af(x)g(x) = af(x)g(x)=af(x)

Horizontal dilation by a factor of 1/b:

g(x)=f(bx)g(x) = f(bx)g(x)=f(bx)

Reflection over the x-axis:

g(x)=−f(x)g(x) = -f(x)g(x)=−f(x)

Reflection over the y-axis:

g(x)=f(−x)g(x) = f(-x)g(x)=f(−x)

General form of combined transformations:

g(x)=acdotf(bx+h)+kg(x) = a cdot f(bx + h) + kg(x)=acdotf(bx+h)+k

How to represent a vertical stretch by a factor of 3?

g(x)=3f(x)g(x) = 3f(x)g(x)=3f(x)

How to represent a horizontal compression by a factor of 2?

g(x)=f(2x)g(x) = f(2x)g(x)=f(2x)

Formula for shifting a function 5 units to the right?

g(x)=f(x−5)g(x) = f(x-5)g(x)=f(x−5)