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  2. AP Pre Calculus
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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.

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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∣<10 < |a| < 10<∣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∣<10 < |b| < 10<∣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.

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).

Given f(x)=x2f(x) = x^2f(x)=x2, find g(x)g(x)g(x) after shifting f(x)f(x)f(x) right 2 units and up 3 units.

  1. Horizontal shift: f(x−2)=(x−2)2f(x-2) = (x-2)^2f(x−2)=(x−2)2. 2. Vertical shift: g(x)=(x−2)2+3g(x) = (x-2)^2 + 3g(x)=(x−2)2+3.

Given f(x)=∣x∣f(x) = |x|f(x)=∣x∣, find g(x)g(x)g(x) after reflecting f(x)f(x)f(x) over the x-axis and stretching vertically by 2.

  1. Reflection: −f(x)=−∣x∣-f(x) = -|x|−f(x)=−∣x∣. 2. Vertical stretch: g(x)=−2∣x∣g(x) = -2|x|g(x)=−2∣x∣.

How to find the equation after reflecting over the y-axis and shifting left by 1?

  1. Reflection: f(−x)f(-x)f(−x). 2. Horizontal shift: g(x)=f(−(x+1))=f(−x−1)g(x) = f(-(x+1)) = f(-x-1)g(x)=f(−(x+1))=f(−x−1).

Describe the steps to transform f(x)f(x)f(x) to 2f(x−1)+32f(x-1) + 32f(x−1)+3.

  1. Shift right 1 unit: f(x−1)f(x-1)f(x−1). 2. Vertical stretch by 2: 2f(x−1)2f(x-1)2f(x−1). 3. Shift up 3 units: 2f(x−1)+32f(x-1) + 32f(x−1)+3.

How do you determine the transformations from f(x)f(x)f(x) to g(x)=f(2x)−1g(x) = f(2x) - 1g(x)=f(2x)−1?

  1. Horizontal compression by a factor of 2. 2. Vertical shift down by 1 unit.

If f(x)=sqrtxf(x) = sqrt{x}f(x)=sqrtx, what is the equation after a horizontal stretch by 3?

g(x)=13xg(x) = \sqrt{\frac{1}{3}x}g(x)=31​x​

Given f(x)=x3f(x) = x^3f(x)=x3, find the equation after reflection over the x-axis and shift down by 4.

  1. Reflection: −f(x)=−x3-f(x) = -x^3−f(x)=−x3. 2. Vertical shift: g(x)=−x3−4g(x) = -x^3 - 4g(x)=−x3−4.

How to transform f(x)f(x)f(x) to g(x)=−f(x+2)−1g(x) = -f(x + 2) - 1g(x)=−f(x+2)−1?

  1. Shift left 2 units: f(x+2)f(x+2)f(x+2). 2. Reflect over x-axis: −f(x+2)-f(x+2)−f(x+2). 3. Shift down 1 unit: −f(x+2)−1-f(x+2) - 1−f(x+2)−1.

Describe the effect of g(x)=f(−x)+2g(x) = f(-x) + 2g(x)=f(−x)+2 on the graph of f(x)f(x)f(x).

  1. Reflect over the y-axis. 2. Shift up by 2 units.

If f(x)=∣x∣f(x) = |x|f(x)=∣x∣, find the equation after a vertical compression by a factor of 0.5 and a shift up by 2 units.

  1. Vertical compression: 0.5∣x∣0.5|x|0.5∣x∣. 2. Vertical shift: g(x)=0.5∣x∣+2g(x) = 0.5|x| + 2g(x)=0.5∣x∣+2.