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.
Vertical translation by k units:
g(x)=f(x)+k
Horizontal translation by h units:
g(x)=f(x+h)
Vertical dilation by a factor of a:
g(x)=af(x)
Horizontal dilation by a factor of 1/b:
g(x)=f(bx)
Reflection over the x-axis:
g(x)=−f(x)
Reflection over the y-axis:
g(x)=f(−x)
General form of combined transformations:
g(x)=acdotf(bx+h)+k
How to represent a vertical stretch by a factor of 3?
g(x)=3f(x)
How to represent a horizontal compression by a factor of 2?
g(x)=f(2x)
Formula for shifting a function 5 units to the right?