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.