Quadratic Functions and Equations—Part 2

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					Algebra 1, Mr. Normile                        November 30, 2009                   Handout Number ___________
Quadratic Functions and Equations—Part 2                                                                 P. 1




                 Quadratic Functions and Equations—Part 2
Review
Here are some of the things we have learned about quadratic functions so far (fill in the blanks):
•   The function formula will always contain an _______________ term.
•   The graph will always be shaped like the letter __________________
•   The rate of change (is always the same or changes): __________________
•   The point where the graph “turns around” is called the _________________________

Making tables and graphs of quadratic functions
In the handout “Quadratic Functions and Equations—Part 1” you made an input-output table and graph
for the function f (x) = x 2 . In this assignment, you will make tables and graphs for more quadratic
functions.
1. Make an input-output table and a graph for the function f(x) = x2 – 5. For your inputs, include some
   negative numbers, zero, and some positive numbers.
     €
           x            f(x)
Algebra 1, Mr. Normile                        November 30, 2009                     Handout Number ___________
Quadratic Functions and Equations—Part 2                                                                   P. 2


2. Make an input-output table and a graph for the function f(x) = –2 · x2. For your inputs, include some
   negative numbers, zero, and some positive numbers.
           x            f(x)




Key vocabulary and facts
Quadratic Function
    A quadratic function is any function whose formula has the form f(x) = ax2 + bx + c.
    (The bx and c terms are optional; that is, b or c could just be 0.)
Parabola
    The graph of a quadratic function is always shaped like ∪ or ∩, and is called a parabola.
Shape of the Parabola
    The shape will be ∪ if a is positive. The shape will be ∩ if a is negative.
Increasing/Decreasing
    If the graph is shaped like ∪, the graph switches from decreasing to increasing at the lowest point.
    If the graph is shaped like ∩, the graph switches from increasing to decreasing at the highest point.
Vertex
    The lowest point of the ∪ or the highest point of the ∩ is called the vertex.
    We say that the function turns around at the vertex.
Axis of Symmetry
    The vertical line through the vertex is called the axis of symmetry. The part of the parabola on one
    side of this line is a mirror image of the part on the other side.
Algebra 1, Mr. Normile                       November 30, 2009                    Handout Number ___________
Quadratic Functions and Equations—Part 2                                                                 P. 3


Another table and graph
3. Answer these questions about the function f(x) = 1x2 – 2x.
    a. Just from looking at the numbers in the function formula, which of the parabola shapes (∪ or ∩)
       will the graph of this function have? Tell how you know.




    b. Make an input-output table and a graph for the function.
       x                f(x)




    c. Find the vertex (highest or lowest point). Circle that point on both the graph and the table.
Algebra 1, Mr. Normile                       November 30, 2009                  Handout Number ___________
Quadratic Functions and Equations—Part 2                                                               P. 4



Application problem
Many things in everyday life and in nature are described by quadratic functions and have parabola shapes,
such as in the following problem.
4. Kim throws a ball out a second-floor window. It flies through the air then lands on the ground. Let x =
   the time since she threw the ball, in seconds.
   Let f(x) = the height of the ball above the ground, in feet.
    The function formula is f(x) = –16x2 + 24x + 16.
    a. Evaluate f(0). What is the meaning of the answer in the problem situation?




    b. Complete the input-output table given below (you may use your calculator to help; for example,
       you could ask it to calculate -16*0.252 + 24*0.25 + 16 ).
       Then make a graph (the grid shown goes from 0 to 5 on the x-axis, 0 to 30 on the y-axis).
       x                f(x)

       0
     0.25
     0.5
     0.75
       1
     1.25
     1.5
     1.75
       2




    c. What was the highest height reached by the ball?


    d. For how much time did the ball stay in the air?