Understanding the Difference Quotient

This video tutorial explains how to determine the difference quotient for a quadratic function. It begins with substituting X + H into the function, followed by expanding and simplifying the expression. The tutorial concludes with the final simplification of the difference quotient, resulting in a simplified expression. The process involves algebraic manipulation, including distributing terms and combining like terms.

What is the first step in determining the difference quotient for a quadratic function?

• Substitute x + h into the function

• Expand the quadratic expression

• Factor the quadratic expression

• Solve for x

When substituting x + h into the function, what should be done to keep the expression organized?

• Use square brackets

• Use curly braces

• Use parentheses

• Use brackets

What is the result of expanding (x + h)^2?

• x^2 + 2x + h^2

• x^2 + 2xh

• x^2 + 2xh + h^2

• x^2 + h^2

What is the purpose of subtracting f(x) from the expanded expression?

• To eliminate x terms

• To simplify the expression

• To factor the expression

• To find the derivative

What is the role of the constant 3 in the expression 3(x + h)^2?

• It changes the axis of symmetry

• It reflects the graph

• It shifts the graph vertically

• It scales the quadratic term

What does the term 6hx represent in the expanded expression?

• A constant term

• A cubic term

• A quadratic term

• A linear term

What happens to the terms in the numerator after clearing the parentheses?

• They are squared

• They are divided by h

• They are combined and simplified

• They are multiplied by h

Why is it important to factor out h from the numerator?

• To simplify the expression

• To eliminate h

• To find the derivative

• To solve for x

What is the simplified form of the difference quotient?

• 5x + 2

• x^2 + 2xh + h^2

• 3x^2 + 6hx + 3h^2

• 6x + 3h - 5

What is the significance of the difference quotient in calculus?

• It is used to solve quadratic equations

• It is used to find the roots of a function

• It determines the maximum value of a function

• It helps in finding the slope of a tangent line