Average Rate of Change of Quadratic Functions

Front

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

Back

The average rate of change of a function over an interval [a, b] is calculated as (f(b) - f(a)) / (b - a).

Front

How do you find the average rate of change of the function b(x) = -2x^2 + 5x - 1 over the interval [1, 3]?

Back

Calculate b(3) = -2(3)^2 + 5(3) - 1 = 3 and b(1) = -2(1)^2 + 5(1) - 1 = 2. Then, average rate of change = (3 - 2) / (3 - 1) = 3.

Front

What is the average rate of change of the function n(x) = 4x^2 - 2x + 1 over the interval [-1, 3]?

Back

Calculate n(3) = 4(3)^2 - 2(3) + 1 = 14 and n(-1) = 4(-1)^2 - 2(-1) + 1 = 5. Then, average rate of change = (14 - 5) / (3 - (-1)) = 14.

Front

Determine the average rate of change of the function p(x) = -x^2 + 3x - 2 over the interval [1, 4].

Back

Calculate p(4) = -4^2 + 3(4) - 2 = 4 and p(1) = -1^2 + 3(1) - 2 = 0. Then, average rate of change = (4 - 0) / (4 - 1) = 4.

Front

Find the average rate of change of the function s(x) = -3x^2 + 6x - 4 over the interval [0, 5].

Back

Calculate s(5) = -3(5)^2 + 6(5) - 4 = 6 and s(0) = -3(0)^2 + 6(0) - 4 = -4. Then, average rate of change = (6 - (-4)) / (5 - 0) = 6.

Front

What is the average rate of change of the function q(x) = 2x^2 - 4x + 7 over the interval [2, 6]?

Back

Calculate q(6) = 2(6)^2 - 4(6) + 7 = 8 and q(2) = 2(2)^2 - 4(2) + 7 = 3. Then, average rate of change = (8 - 3) / (6 - 2) = 8.

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 graph of a quadratic function look like?

Back

The graph of a quadratic function is a parabola, which opens upwards if a > 0 and downwards if a < 0.

Front

What is the vertex of a quadratic function?

Back

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction the parabola opens.

Front

How do you find the vertex of a quadratic function in standard form?

Back

The vertex (h, k) can be found using the formula h = -b/(2a) and k = f(h), where f(x) is the quadratic function.