Midterm Exam

The range of a function is the set of all possible output values (y-values) that the function can produce based on its domain (input values).

To reflect a graph across the y-axis, replace every x in the function with -x. For example, the reflection of y = |x - 1| + 7 is y = |x + 1| + 7.

The solutions are x = 3 and x = -4.

A parent function is the simplest form of a function type, from which other functions can be derived by transformations such as translations, reflections, and stretches.

Translating a function means shifting its graph horizontally and/or vertically without changing its shape.

Adding a constant to a function shifts the graph vertically. For example, f(x) = |x| + 2 shifts the graph of f(x) = |x| up by 2 units.

The notation |x| represents the absolute value of x, which is the distance of x from 0 on the number line, always resulting in a non-negative value.

For a quadratic function in the form y = ax² + bx + c, the vertex can be found using the formula x = -b/(2a).

A closed interval includes its endpoints (e.g., [a, b]), while an open interval does not include its endpoints (e.g., (a, b)).

The y-intercept is the point where the graph of the function crosses the y-axis, representing the output value when the input value (x) is zero.