Understanding Quadratic Functions and Their Zeros

The video tutorial explains how to solve the quadratic function f(x) = -x^2 + 4x by finding the x-values for which f(x) equals zero. The process involves substituting zero for f(x), rearranging the equation, and factoring it to find the solutions x = 0 and x = 4. The solutions are verified by graphing the function and observing the points of intersection with the x-axis.

What is the function given in the problem that needs to be solved for f(x) = 0?

• f(x) = x^2 + 4x

• f(x) = -x^2 - 4x

• f(x) = -x^2 + 4x

• f(x) = x^2 - 4x

What is the first step in solving the equation f(x) = 0 for the given function?

• Graph the function

• Multiply by negative one

• Substitute zero for x

• Substitute zero for f(x)

Why do we multiply both sides of the equation by negative one?

• To simplify the equation

• To make the leading term positive

• To eliminate the x term

• To make the equation more complex

What is the greatest common factor that is factored out from the equation?

• x^2

• 4

• x

• -x

What property is used to solve the equation after factoring?

• Associative Property

• Commutative Property

• Zero Product Property

• Distributive Property

What are the solutions for x when the equation is solved?

• x = 0 and x = -4

• x = 1 and x = -1

• x = 0 and x = 4

• x = 2 and x = -2

What does the graph of f(x) = -x^2 + 4x intersect with to verify the solutions?

• The y-axis

• The line y = x

• The x-axis

• The line y = -x

What is the ordered pair for the point of intersection at the origin?

• (4, 0)

• (4, 4)

• (0, 0)

• (0, 4)

What does the ordered pair (4, 0) represent in the context of the graph?

• When x is 4, f(x) is 4

• When x is 0, f(x) is 4

• When x is 4, f(x) is 0

• When x is 0, f(x) is 0

How does the graph verify the solutions found algebraically?

• By not intersecting the x-axis

• By showing intersections at the solutions

• By showing no intersections

• By showing intersections at different points