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 $f(x) + k$ and $f(x + k)$ .

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

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

$af(x)$ : Vertical dilation by a. $f(ax)$ : 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 $g(x) = af(x)$ .

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

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 $f(x) + k$ .

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

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

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

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

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

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

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

Why does horizontal translation appear 'opposite'?

Because $f(x+h)$ 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 $g(x) = af(x)$ .

It reflects the graph of $f(x)$ 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.

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

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

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

Reflection: $-f(x) = -|x|$ . 2. Vertical stretch: $g(x) = -2|x|$ .

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

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

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

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

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

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

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

$g(x) = \sqrt{\frac{1}{3}x}$

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

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

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

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

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

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

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

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

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