Average Rate of Change (Quadratic)

Front

What is the average rate of change of a function between two points?

Back

The average rate of change of a function between two points (x1, f(x1)) and (x2, f(x2)) is calculated as: \( \frac{f(x2) - f(x1)}{x2 - x1} \).

Front

How do you find the slope of a line segment on a graph?

Back

The slope (m) of a line segment between two points (x1, y1) and (x2, y2) is given by: \( m = \frac{y2 - y1}{x2 - x1} \).

Front

What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically in the form: \( f(x) = ax^2 + bx + c \), where a, b, and c are constants and a ≠ 0.

Front

What does the term 'interval' mean in the context of functions?

Back

An interval refers to a range of x-values over which a function is analyzed, often denoted as [a, b] or (a, b).

Front

How do you calculate the average rate of change for the function f(x) = 2x^2 + 12x + 16 on the interval [-3, -2]?

Back

To find the average rate of change, calculate: \( \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{(2(-2)^2 + 12(-2) + 16) - (2(-3)^2 + 12(-3) + 16)}{-2 + 3} = 2 \).

Front

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

Back

The slope indicates the rate of change of the function at a given point. In a quadratic function, the slope changes as x changes, reflecting the curve's shape.

Front

What does it mean if the average rate of change is zero?

Back

If the average rate of change is zero over an interval, it means that the function's values at the endpoints of the interval are equal, indicating a horizontal line segment.

Front

How can you determine if a rate of change is increasing or decreasing?

Back

If the average rate of change between two intervals is positive, the function is increasing; if negative, it is decreasing.

Front

What is the formula for the average rate of change of a quadratic function?

Back

The average rate of change of a quadratic function \( f(x) = ax^2 + bx + c \) over the interval [x1, x2] is given by: \( \frac{f(x2) - f(x1)}{x2 - x1} \).

Front

What does it mean if the rate of change is undefined?

Back

An undefined rate of change typically occurs when the denominator of the slope formula is zero, indicating a vertical line or a discontinuity in the function.