Function Transformations and Rates of Change Study Guide

Definition: A graph has reflectional symmetry over the y-axis if it remains unchanged when reflected across the y-axis. This means if (x, y) is on the graph, then (-x, y) is also on the graph.
Examples from provided graphs:
- Quadratic Function: Generally, quadratic functions of the form f(x) = ax^2 have reflectional symmetry over the y-axis. The example graph of a quadratic function does exhibit this symmetry.
- Cubic Function: Functions like f(x) = x^3 do not have reflectional symmetry over the y-axis. They often have point symmetry about the origin.
- Square Root Function: Functions like f(x) = \sqrt{x} do not have reflectional symmetry over the y-axis, as their domain is typically restricted to x \ge 0 or x \le 0.
- Absolute Value Function: Functions like f(x) = |x| do have reflectional symmetry over the y-axis.

Graph Components:
- Y-intercept: The point where the graph crosses the y-axis (where x = 0).
- X-intercepts: The points where the graph crosses the x-axis (where y = 0). A quadratic function can have two, one (a double root), or no x-intercepts.
- Vertex: The highest or lowest point on a parabola. For a parabola opening upwards, the vertex is the minimum. For a parabola opening downwards, it is the maximum.
- Axis of Symmetry: A vertical line that passes through the vertex of a parabola, dividing it into two mirrored halves. Its equation is typically x = h, where (h, k) is the vertex.
- Domain: For all quadratic functions, the domain is all real numbers, denoted as (-\infty, \infty).
- Range: Dependent on the vertex and direction of opening. For an upward-opening parabola with vertex (h, k), the range is [k, \infty). For a downward-opening parabola, the range is (-\infty, k].
- Asymptote: This term is not applicable to quadratic functions. Asymptotes are lines that a graph approaches but never touches.

Context: A ball's height is a function of its horizontal distance.
Data Analysis from Table and Graph:
- Increasing Interval: The segment of the graph where the y-values (height) are rising as x-values (distance) increase. For the ball toss, this is from a distance of 0 feet to 5 feet (0 \le X \le 5).
- Decreasing Interval: The segment of the graph where the y-values (height) are falling as x-values (distance) increase. For the ball toss, this is from a distance of 5 feet to 10 feet (5 \le X \le 10).
- Starting Point: The initial position of the ball ((0, 2), meaning at 0 feet horizontal distance, the ball is 2 feet high).
- Ending Point: The final position recorded for the ball ((10, 2), meaning at 10 feet horizontal distance, the ball is 2 feet high).
- Maximum Height: The highest y-value reached by the ball during its trajectory. In this case, it's 7 feet (occurring at 5 feet horizontal distance).

Relative Maximum: A point on a graph where the function changes from increasing to decreasing. Visually, it's a