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Given $f(x) = x^2$, find $g(x)$ after shifting $f(x)$ right 2 units and up 3 units.
1. 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.
1. 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?
1. Reflection: $f(-x)$. 2. Horizontal shift: $g(x) = f(-(x+1)) = f(-x-1)$.
Describe the steps to transform $f(x)$ to $2f(x-1) + 3$.
1. 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 $g(x) = f(2x) - 1$?
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.
1. Reflection: $-f(x) = -x^3$. 2. Vertical shift: $g(x) = -x^3 - 4$.
How to transform $f(x)$ to $g(x) = -f(x + 2) - 1$?
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 $f(x)$.
1. 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.
1. Vertical compression: $0.5|x|$. 2. Vertical shift: $g(x) = 0.5|x| + 2$.
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.