Average Rate of Change (Quadratic)

Front

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

Back

The average rate of change of a function between two points is the change in the function's value divided by the change in the input value, calculated as \( \frac{f(b) - f(a)}{b - a} \) for the interval [a, b].

Front

How do you calculate the average rate of change for the function \( y = \frac{1}{5}x^2 - 12.8 \) on the interval [3, 8]?

Back

1. Calculate \( f(8) = \frac{1}{5}(8^2) - 12.8 = 0.8 \) and \( f(3) = \frac{1}{5}(3^2) - 12.8 = -11.2 \). 2. Average rate of change = \( \frac{0.8 - (-11.2)}{8 - 3} = \frac{12}{5} = \frac{11}{5} \).

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 \( a \neq 0 \).

Front

What does it mean for a function to have a greater rate of change?

Back

A function has a greater rate of change if its average rate of change over a specific interval is larger than that of another function, indicating it increases or decreases more steeply.

Front

How do you determine the rate of change between two points on a graph?

Back

To determine the rate of change between two points (x1, y1) and (x2, y2), use the formula \( \frac{y2 - y1}{x2 - x1} \).

Front

What is the average rate of change of the function \( g(x) = -x^2 + x + 3 \) on the interval [0, 2]?

Back

1. Calculate \( g(2) = -2^2 + 2 + 3 = 3 \) and \( g(0) = 3 \). 2. Average rate of change = \( \frac{3 - 3}{2 - 0} = 0 \).

Front

What is the significance of the average rate of change in real-world applications?

Back

The average rate of change can represent speed, growth rate, or any change over time, helping to analyze trends and make predictions.

Front

What is the average rate of change of a linear function?

Back

The average rate of change of a linear function is constant and equal to the slope of the line, calculated as \( \frac{y2 - y1}{x2 - x1} \).

Front

How can you use a table to find the average rate of change?

Back

To find the average rate of change using a table, identify the values of the function at two points, then apply the formula \( \frac{f(b) - f(a)}{b - a} \).

Front

What is the average rate of change of the function on the interval [1, 5] if the values are given?

Back

If the function values are f(1) and f(5), use the formula \( \frac{f(5) - f(1)}{5 - 1} \) to calculate the average rate of change.