4 Chapter Test

Do you know HOW?

Sketch a graph of the quadratic function with the given vertex and through the given point. Then write the equation of the parabola in vertex form and describe how the function was transformed from the parent function y = x 2 . y equals , x squared , .

1. vertex (0, 0), point ( − 2 , 3 ) open negative 2 comma 3 close
2. vertex (1, 5), point (2, 1)

Graph each quadratic function. Identify the axis of symmetry, the vertex, and the domain and the range of each function.

3. y = x 2 − 7 y equals , x squared , minus 7
4. y = x 2 + 2 x + 6 y equals , x squared , plus 2 x plus 6
5. y = − x 2 + 5 x − 3 y equals negative , x squared , plus 5 x minus 3

Simplify each expression.

6. − 16 square root of negative 16 end root
7. 4 − 9 − 2 4 , square root of negative 9 end root , minus 2
8. ( 2 + 3 i ) ( 8 − 5 i ) open 2 plus 3 i close open 8 minus 5 i close
9. ( − 3 + 2 i ) − ( 6 + i ) open negative 3 plus 2 i close minus open 6 plus i close
10. 4 + 2 i 2 − i fraction 4 plus 2 i , over 2 minus i end fraction

Factor each expression completely.

11. 2 y 2 − 8 y 2 y squared , minus 8 y
12. 3 x 2 + 8 x − 3 3 x squared , plus 8 x minus 3
13. 9 w 2 − 30 w + 25 9 w squared , minus 30 w plus 25

Solve each quadratic equation.

14. x 2 − 25 = 0 x squared , minus 25 equals 0
15. x 2 − 2 x + 3 = 0 x squared , minus 2 x plus 3 equals 0
16. x 2 − 8 x = − 6 x squared , minus 8 x equals negative 6

Solve the following systems of equations.

17. { y = 3 x 2 − x + 1 y = 3 x 2 + x − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals 3 , x squared , minus x plus 1 , row2 column 1 , y equals 3 , x squared , plus x minus 1 , end table
18. { y = − x 2 + 2 x − 3 y = 4 x − 3 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , plus 2 x minus 3 , row2 column 1 , y equals 4 x minus 3 , end table

Solve the following systems of inequalities.

19. { y > 2 x 2 + 5 x + 1 y < − 2 x 2 − 5 x − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y greater than 2 , x squared , plus 5 x plus 1 , row2 column 1 , y less than negative 2 , x squared , minus 5 x minus 1 , end table
20. { y < x 2 − x + 2 y > x 2 − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y less than , x squared , minus x plus 2 , row2 column 1 , y greater than , x squared , minus 1 , end table

Evaluate the discriminant of each equation. How many real and imaginary solutions does each have?

21. x 2 + 6 x − 7 = 0 x squared , plus 6 x minus 7 equals 0
22. 3 x 2 − x + 3 = 0 3 x squared , minus x plus 3 equals 0
23. − 4 x 2 − 4 x + 1 = 0 negative 4 , x squared , minus 4 x plus 1 equals 0

Do you UNDERSTAND?

24. Writing Compare graphing a number on the complex plane to graphing a point on the coordinate plane. How are they similar? How are they different?
25. Open-Ended Sketch the graph of a quadratic function f ( x ) = a x 2 + b x + c f open x close equals eh , x squared , plus b x plus c  that has no real zeros. How does this relate to the solutions of the related equation a x 2 + b x + c = 0 ? eh , x squared , plus b x plus c equals 0 question mark
26. Physics A model for the path of a toy rocket is given by h = 68 t − 4 . 9 t 2 , h equals 68 , t minus , 4 . 9 , t squared , comma  where h is the altitude in meters and t is the time in seconds. Explain how to find both the maximum altitude of the rocket and how long it takes to reach that altitude.
27. How many solutions are possible for:
    1. a system of two quadratic equations?
    2. a system of two quadratic inequalities? Explain your answers.

Table of Contents