What two values of x make the product above equal to zero?

Plug each of these values into x2 + 3x + 2. What do you get?

The blue points are x-intercepts of the parabola. They are the points where y = 0. Mouseover the blue points. What is the x-coordinate of each point?

Plug each solution into x2 + 3x + 2. What do you get?

Recall that x^2 + 3x + 2 = (x + 1)(x + 2). How do these factors relate to the x-intercepts of y = x^2 + 3x + 2?

The roots of a quadratic equation are the values of x that make the related function zero. The real roots are also the x-intercepts of the parabola. Look at the graph of y = x2 – 4x + 3.

How many roots does x2 – 4x + 3 = 0 have?


What are the roots?




Change c to 4.0. How many roots does x2 – 4x + 4 = 0 have?



Now graph y = x^2 – 4x + 8 in the Gizmo, and look at the resulting parabola. Do you think
x^2 – 4x + 8 = 0 has any real roots? Explain.





Vary the values of a, b, and c. In general, how many real roots are possible for a quadratic equation?

Graph y = x2 + 6x + 5. Turn on Show axis of symmetry x = –b/(2a). The axis of symmetry is a line that divides a parabola into two halves that are mirror images.

How does the location of the axis of symmetry relate to the location of the two
x-intercepts?





Move the a, b, and c sliders. Which values affect the axis of symmetry?



The equation of the axis of symmetry is x = . How does this explain what you observed above?

Suppose you know the line of symmetry for a quadratic function.

From just this information, can you find the x-intercepts?


Explain.






Suppose the axis of symmetry of the graph of a quadratic function is at x = 6. If one root of the related quadratic equation is –1.5, what is the other root?