Dynamic Algebra™ Exploration

Factored Form of Quadratic Equations

On this web page, you can explore the relationships between the factored form of a quadratic equation, the graph of the equation, and the roots when the equation is set equal to zero. This exploration will help you better understand the concepts in Lesson 9.4—Factored Form—on pages 515–521 of Discovering Algebra: An Investigative Approach.

Sketch

This sketch shows the graph of a quadratic equation in factored form:

Drag the sliders, r₁ and r₂, to change the integer values inside the parentheses. Click Start Over to reset the equation to y = (x − 2)(x – 4).

Investigate

1. Set the sketch to the starting equation, y = (x – 2)(x – 4). Describe the graph. Is the graph a parabola? How do you know?
2. Where are the x-intercepts of the graph of y = (x – 2)(x – 4)?
3. Find the roots of the equation 0 = (x – 2)(x – 4).
4. How are the roots of the equation 0 = (x – 2)(x – 4) related to the x-intercepts of the graph y = (x – 2)(x – 4)?
5. Change the equation by dragging the sliders. Is the relationship you observed in Question 4 still true? If the sketch allowed you to set r₁ and r₂ to any real value, would the relationship still be true? Explain why or why not.
6. In the sketch, what happens to the equation when either r₁ or r₂ becomes negative? How would you normally write an equation like this?
7. When both roots are positive, what happens to the graph? What happens when both roots are negative? What happens when both roots are equal?
8. How is the vertex of the parabola related to the x-intercepts?
9. How would your answer to Question 8 change if you were to graph a quadratic equation in factored form that has a leading coefficient, y = a(x – r₁)(x – r₂)?

Summarize

1. How are the roots of a quadratic equation related to the x-intercepts of the corresponding function?
2. Why does this relationship hold?
3. How can you use this relationship in your study of quadratics?