How does a vertical stretch affect the steepness of a graph?
It increases the steepness if the stretch factor is greater than 1 and decreases it if the factor is between 0 and 1.
How does a horizontal shift affect the x-intercepts of a graph?
It shifts the x-intercepts by the same amount as the horizontal shift. A shift to the right increases the x-intercept values, and a shift to the left decreases them.
What does a reflection over the x-axis do to the y-values on a graph?
It changes the sign of all y-values, flipping the graph over the x-axis.
How does a horizontal compression affect the period of a periodic function?
It decreases the period by the compression factor, making the function oscillate more rapidly.
What does the graph of y=f(−x) look like compared to y=f(x)?
It's a reflection of y=f(x) over the y-axis.
How can you identify a vertical translation from a graph?
The entire graph is shifted up or down without changing its shape.
What happens to the vertex of a parabola after a horizontal shift?
The x-coordinate of the vertex shifts by the same amount as the horizontal shift.
How does a vertical compression affect the maximum and minimum values of a function?
It reduces the maximum and minimum values by the compression factor.
What does a reflection over the y-axis do to a function's symmetry?
If the original function was even (symmetric about the y-axis), the reflection doesn't change the graph. If it was odd (symmetric about the origin), the reflection changes the sign of the function.
How does a horizontal stretch affect the domain of a function?
It expands the domain by the stretch factor.
What are the differences between vertical and horizontal translations?
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).