Understanding Functions and Graphs

Front

What is the average rate of change of a function over an interval?

Back

The average rate of change of a function f(x) over the interval [a, b] is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

Front

How do you find the vertex of a parabola given in vertex form?

Back

The vertex of a parabola given by the equation \( y = a(x - h)^2 + k \) is the point (h, k). This form makes it easy to identify the vertex directly.

Front

What is function composition?

Back

Function composition is the process of applying one function to the results of another function. If f(x) and g(x) are two functions, the composition f(g(x)) means to apply g first and then apply f to the result.

Front

What is the domain of the function \( y = \sqrt{x} \)?

Back

The domain of the function \( y = \sqrt{x} \) is all real numbers x such that \( x \geq 0 \). This is because the square root function is only defined for non-negative values.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is \( y = ax^2 + bx + c \), where a, b, and c are constants and a ≠ 0.

Front

How do you determine if a function is increasing or decreasing?

Back

A function is increasing on an interval if its derivative is positive on that interval, and it is decreasing if its derivative is negative.

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex of a quadratic function represents the maximum or minimum point of the parabola, depending on the direction it opens (upward or downward).

Front

What is the difference between a function and a relation?

Back

A function is a specific type of relation where each input (x-value) is associated with exactly one output (y-value). In contrast, a relation can have multiple outputs for a single input.

Front

What is the range of a quadratic function in vertex form?

Back

For a quadratic function in vertex form \( y = a(x - h)^2 + k \), the range depends on the value of a. If a > 0, the range is \( [k, \infty) \); if a < 0, the range is \( (-\infty, k] \).

Front

How do you find the x-intercepts of a quadratic function?

Back

To find the x-intercepts of a quadratic function, set the function equal to zero and solve for x. This can be done using factoring, completing the square, or the quadratic formula.