Math 2 Honors Name___________________________Lesson 3-3: Designing Parabolas Date__________________________

Learning Goals: I can identify which parts of the function indicate, if applicable, the function’s y-intercept, x-intercept(s),

increasing intervals, decreasing intervals, minimums, maximums, symmetries, end behaviors. I can apply the zero product property to find the zeros of a quadratic function written in factored form. I can write the equation that describes a quadratic function in factored form when I am given a graph with the x-

intercept(s) and another point on the graph.

Recall from last unit, the coefficient of the x2 term, tells us a great deal about the parabola..If f(x) = k · x2, then answer the following questions:

1. If k is positive, then which way does the parabola open? _____________2. If k is negative, then which way does the parabola open? _____________

3. If |k| > 1, then is the parabola vertically stretched or compressed? _____________

4. If 0 < |k| < 1, then is the parabola vertically stretched or compressed? _____________

Next, you are going to find the equation of the parabola going through the path of Math-O, the Daredevil, by applying what you reviewed up above, plus a little bit more . . . . .

NOTES:

1. Does the parabola have a maximum or a minimum? _________________ Based on your answer, what is the sign of k? ________________

2. Compared to the parent function, f(x) = x2 (also pictured in the graph above), is Matho’s path a stretch or compression? _____________ Based on your answer, what are some possible values of k? _________________________

3. What are the x-intercepts of the parabola? ( , 0) and ( , 0)Write a pair of linear factors that would yield those x-intercepts: ( x )( x )

4. Now use your answers from questions 1, 2, & 3 to create an equation for Matho’s path. Enter your equation in the calculator to test it out.

Describe how well your parabola fits Matho’s path:

What adjustment(s) to your equation are you going to make to get a better fit?

Write your final equation in factored form: f(x) = ____________________________ OVER

Page 2Write an equation for the parabola in the picture below:

This time do not guess & check. Set up an equation and solve for k. Show your work below:

NOTES:

3.

Use reasoning alone to sketch graphs of the following functions. Label these key points with their coordinates on the graphs:

x-intercept(s), y-intercept, and maximum or minimum point.

4a.

4b. 4c.

4d. 4e.

Write rules for quadratic functions whose graphs have the following properties. If possible, write more than one function rule that meets the given conditions.

5a. x-intercepts at (4, 0) and (-1, 0)

5b. x-intercepts at (7, 0) and (1, 0) and graph opening upward

5c. x-intercepts at (7, 0) and (1, 0) and a minimum point at (4, -10)

5d. x-intercepts at (-5, 0) and (0, 0) and graph opening downward

5e. x-intercepts at (3, 0) and (-5, 0) and maximum point at (-1, 8)

5f. x-intercepts at (3.5, 0) and (0, 0) and graph opening upward

5g. x-intercepts at (4.5, 0) and (1, 0) and y-intercept at (0, 9)

5h. x-intercepts at (m, 0) and (n, 0)

5i. only one x-intercept at (0, 0)

5j. only one x-intercept at (2, 0) and y-intercept at (0, 6)