All Flashcards
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).
Given , find after shifting right 2 units and up 3 units.
- Horizontal shift: . 2. Vertical shift: .
Given , find after reflecting over the x-axis and stretching vertically by 2.
- Reflection: . 2. Vertical stretch: .
How to find the equation after reflecting over the y-axis and shifting left by 1?
- Reflection: . 2. Horizontal shift: .
Describe the steps to transform to .
- Shift right 1 unit: . 2. Vertical stretch by 2: . 3. Shift up 3 units: .
How do you determine the transformations from to ?
- Horizontal compression by a factor of 2. 2. Vertical shift down by 1 unit.
If , what is the equation after a horizontal stretch by 3?
Given , find the equation after reflection over the x-axis and shift down by 4.
- Reflection: . 2. Vertical shift: .
How to transform to ?
- Shift left 2 units: . 2. Reflect over x-axis: . 3. Shift down 1 unit: .
Describe the effect of on the graph of .
- Reflect over the y-axis. 2. Shift up by 2 units.
If , find the equation after a vertical compression by a factor of 0.5 and a shift up by 2 units.
- Vertical compression: . 2. Vertical shift: .
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.