All Flashcards
What is a vertical translation?
Shifting a function's graph up or down by adding/subtracting a constant.
What is a horizontal translation?
Shifting a function's graph left or right by adding/subtracting a constant to the input.
What is a vertical dilation?
Stretching or shrinking a function's graph vertically by multiplying the function by a constant.
What is a horizontal dilation?
Stretching or shrinking a function's graph horizontally by multiplying the input by a constant.
What is a reflection over the x-axis?
Flipping a function's graph over the x-axis, achieved by multiplying the function by -1.
What is a reflection over the y-axis?
Flipping a function's graph over the y-axis, achieved by replacing x with -x.
Define transformation of a function.
Altering the graph of a function by shifting, stretching, shrinking, or reflecting it.
What does 'dilation' mean in function transformations?
Stretching or compressing the graph of a function either vertically or horizontally.
What is the general form equation for combined transformations?
Define vertical stretch.
A transformation that multiplies all y-values of a function by a factor greater than 1, making the graph taller.
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 look like compared to ?
It's a reflection of 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?
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 and .
: Vertical shift by k. : Horizontal shift by -k.
Compare the effects of and .
: Vertical dilation by a. : 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 .
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 affect the graph differently than changing 'a' in ?
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).