Identify Functions & Transformations
State whether the relation { (1, 2), (3, 4), (5, 4), (7, 8) } is a function. Explain why or why not.
Yes, because every x-value corresponds to exactly one y-value (no x-value repeats with a different y-value)
State the domain of any standard quadratic function
y = a(x - h)2 + k
without any real-world restrictions.
{ xER}
State the vertex and equation of the axis of symmetry for y = 3(x - 4)2 + 11
Vertex is (4, 11); Axis of symmetry is x = 4.
Write the general mapping notation formula (x, y) -> (..., ...)y = a(x - h)2 + k
(x, y) -> (x + h, ay + k)
The height of a soccer ball in meters is h(t) = -5(t - 2)2 + 20. What is the maximum height reached by the ball?
20 meters (the k-value of the vertex).
Identify the transformations applied to y = x2 to obtain y = -4(x + 2)2
Reflection in the x-axis, vertical stretch by a factor of 4, and horizontal translation 2 units left.
State the domain and range for y = (x - 3)2 + 4
Domain: { xER}
Range: { yER | y ≥ 4 }
For y = -0.5(x + 2)2 - 8, state whether the function has a maximum or minimum value, and state what that optimal value is.
Maximum value of -8 (occurs at x = -2).
Apply the mapping rule for y = -(x + 4)2 - 3 to the base points (-1, 1), (0, 0), and (1, 1).
Mapping rule: (x, y) -> (x - 4, -y - 3)
(-1, 1) -> (-5, -4)
(0, 0) -> (-4, -3)
(1, 1) -> (-3, -4)
A company's profit is modeled by P(x) = -2(x - 50)2 + 5000$, where x is items sold. How many items maximize profit?
50 items (the h-value of the vertex).
Identify the transformations applied to y = x2 to obtain y = -4(x - 3)2 + 6
Reflection in the x-axis, vertical stretch by a factor of 4, vertical translation 6 units up and horizontal translation 3 units right.
A quadratic function opens downwards and has its vertex at (-2, 7). State its domain and range in set notation.
Domain: { xER}
Range: { yER | y ≤ 7 }
How do you find the y-intercept of a quadratic function directly from its vertex form equation y = 2(x - 3)2 - 8? Calculate it.
Set x = 0 and solve for y: y = 2(0 - 3)2 - 8. The y-intercept is (0, 10)
Given the mapping rule (x, y) -> (x - 5, -3y + 2), write the transformed vertex form equation and state its vertex.
Equation: y = -3(x + 5)2 + 2; Vertex: (-5, 2).
A stone's height in meters after t seconds is h(t) = -5(t - 3)2 + 45. At what time (t) does the stone hit the ground?
0 = -5(t - 3)2 + 45
t = 6 seconds.