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#### S. Kastberg & T. Barron

X-INTERCEPTS FOR FUNCTIONS OF THE FORM 1. Use a computer graphing tool to sketch the function and estimate the x-intercepts.

2. To find the x-intercepts algebraically, we substitute for and solve for .

3. Use what you recalled in problem two to set up an equation that ,when solved, will tell you the x-intercepts. You can use the following notation for , y = (x^2) - 4.

4. Solve the equation that you wrote in problem 3 for .

5. Now solve for x in the equation above, what operation must be used? (What is the inverse operation of squaring?)

6. Find the x-intercepts by applying the operation you discovered in problem 5 to the equation you wrote in problem 4. Keep in mind that there are two x-intercepts in this case. .

7. Use a computer graphing tool to sketch the function and estimate the x-intercepts.

8. Write an equation that if solved will find the x-intercepts.

9. Solve the equation in problem 8 for and use the operation you discovered in problem 5 to solve for x . (Remember there should be two x-intercepts.)

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10. Explain in complete sentences using the method sketched above how to find the x-intercepts for the function .

11. Explain in complete sentences using the method sketched above how to find the x-intercepts for the function . Note, division by 2 is usually done before you take the square root.

CONVERTING FROM FORM TO FORM

12. Suppose we are asked to find the graph and find the x-intercepts of the function . To use the method we developed in the last lesson we must convert this equation to the form .

13. Suppose that we rewrite in the following form:

f(x) = ( - 2x + ______ ) + 3 - ______

What is the smallest number that can be written in the blanks above so that ( - 2x + ______ ) can be factored into (x - _____ )^2 ? .

14. So f(x) = ( - 2x + ) + 3 - = .

15. Consider . Fill in the following blanks with a number so that + + 3 - can be factored and simplified.

16. Consider . Fill in the following blanks with a number so that + + 3 - can be factored and simplified.

17. Do you notice a relationship between the coefficient of x and the number you find? What is the relationship?

18. Rewrite in form.

19. Congratulations you just found the vertex of the function and you have learned how to complete the square!

Now find the x-intercepts for the function .

20. Now what if you had a quadratic like this one .

BEWARE!

The coefficient of the squared term, 2, causes problems when we complete the square!

After we rewrite the function in the form + ______ + 1 - __________, we must factor a two out of the parentheses.

Factoring out the two leaves f(x) = 2( + _________ ) + 1 - _______.

21. Now we can fill in blanks above with the appropriate value, however the second blank must be filled in a with times the value entered in the first blank. Explain why you think this is necessary.

22. Finish the factorization of .

23. If you were faced with the function , we would rewrite the function being careful to factor out -5 from the first two terms. So the factored form of the function would be .

24. Find the vertex of by completing the square. Explain what you did in complete sentences.

25. But there has to be an easier way to find the vertex and the x-intercepts of a quadratic in the form . Well, there is!

Consider the function . Us the method we learned above to find the vertex and the x-intercepts. A, b, c are all real numbers, so do not be intimidated. I will get you started, fill in the blanks below to find the function in the form .

f(x) = a( + ) + c +a ( )

26. Now you know the vertex of the function is .

27. Find the x-intercept by solving the equation that you found in problem 25.

Explain what you did in complete sentences.

Congratulations you just discovered the quadratic formula!

28. What types of equations do you think this equation solves? .

29. Use the formulas you found to find the vertex and the x-intercepts of the following function Vertex = x-intercepts = .

30. Use a computer graphing tool to graph .

The domain of the function is .

The range of the function is

The interval over which the function is decreasing is .

The interval over which the function is increasing is .

The vertex of the parabola is

The x-intercepts of the parabola are .