chapter 4 math
A point (2, 5) on y = f(x) is translated to (5, 6) on the transformed graph. What are the values of h and k?
h = 3, k = 1
If y = f(x) is transformed to y + 1 = 2f(x - 3), how is the point (0,0) moved?
(3, -1)
If b = 1/2 in the equation y = af(bx), what is the factor of horizontal stretch or compression?
A horizontal stretch by a factor of 2
If a point (-3, 8) is on y = f(x), what point is on its inverse, y = f⁻¹(x)?
(8, -3)
What is the general equation form for a combination of transformations?
y - k = af(b(x - h))
Which algebraic step is the first one when finding the inverse of a function?
Interchange x and y
If a function's graph is shifted 3 units down, how does this affect the 'k' value in the equation y - k = f(x - h)?
k = -3
A horizontal compression by a factor of 1/3 means the 'b' value in y = af(bx) is what?
b = 3
What is the core characteristic of an inverse function relative to its original function?
It reverses the action of the original function.
What is the general order of operations for applying transformations to a point (x, y) according to the combined point transformation rule (x/b + h, ay + k)?
Horizontal stretch/compression/reflection, then vertical stretch/compression/reflection, then horizontal translation, then vertical translation.
When finding the inverse of a function algebraically, what do you solve for after interchanging x and y?
y
What is the equation form for a function reflected in the y-axis?
y = f(-x)
What type of transformation is defined as shifting the graph of a function up/down or left/right without changing its shape?
Vertical and/or Horizontal Translation
Which transformation type affects the 'y' coordinate by multiplying it by 'a' in the combined point rule?
Vertical stretch/compression/reflection
Describe the transformation caused by 'b' if 0 < b < 1 in the general equation form y - k = af(b(x - h)).
Horizontal stretch
If f(x) = 2x + 1, what is f⁻¹(x)?
f⁻¹(x) = (x - 1) / 2
According to the combined point transformation rule, which parameter primarily affects the vertical stretch or compression of a function?
a
A point (1, 1) on y = f(x) is transformed by y - 2 = 3f(2(x + 1)). What are the new coordinates?
(-0.5, 5)
A point (6, 9) on y = f(x) is transformed to (3, 18) by y = af(bx). What are the values of 'a' and 'b'?
a = 2, b = 2
How does a reflection in the x-axis transform a point (x, y) on the original function?
(x, -y)
What does 'h' represent in the general equation form y - k = af(b(x - h))?
Horizontal translation
In y = af(bx), what effect does 'a' have if a > 1?
Vertical stretch
The graph of an inverse function is a reflection of the original function across which line?
y = x
What does a positive value of 'h' represent in the equation y - k = f(x - h)?
Horizontal translation right
A point (4, -7) is reflected in the x-axis. What are the new coordinates of the point?
(4, 7)