1. Review 2.1 and Checkpoint

2. List of Topics to Know Quadratic Transformations ◦Stretch/Compress ◦Shift Left and Right ◦Shift Up and Down ◦Flip Up and Down ◦Finding the Vertex Graphing Form vs. Standard Form ◦Completing the Square ◦Averaging the Intercepts Sketching a Parabola from Graphing Form ◦Over 1 and 2, Up 1 and 4 (Stretch Factor??) Modeling with Parabolas Distance Between 2 Points (CP) Writing Equation of a Line Given 2 Points (CP) Factoring Simplifying Radicals ◦Prime Factorization **REMINDER: NO CALCULATOR!!!!

3. Forms of Parabolas Graphing Form (or Vertex Form): ◦y = a(x – h) 2 + k ◦Vertex is (h, k) Standard Form: ◦y = ax 2 + bx + c Factored Form: ◦y = a(x + b)(x + c)

4. Converting Standard Form to Graphing Form AVERAGING THE INTERCEPTS y = x 2 – 2x – 24 Factor using “box and diamond” 0 = (x – 6)(x + 4) Find/Average the x-intercepts x = 6 and -4 Average is 1 Substitute x into the equation to solve for y y = 1 2 – 2(1) – 24 = -25 Vertex is (1, -25) Write in graphing form using the vertex you calculated and the value of a from original form y = (x – 1) 2 – 25 COMPLETING THE SQUARE y = x 2 – 2x – 24 Move constant to other side to create the blank and fill blank(s) with (half of middle term) 2 y + 24 + 1 = x 2 – 2x + 1 Simplify both sides y + 25 = (x – 1) 2 Solve for y y = (x – 1) 2 – 25

5. Example Problem Rewrite the following equation in graphing form by averaging the intercepts OR by completing the square. y = x 2 – 12x + 35 ANSWER: y = (x – 6) 2 – 1

6. Quadratic Transformations y = a(x – h) 2 + k Vertex (h, k) “h” – shifts parabola left and right “k” – shifts parabola up and down Sign of “a” determines orientation (opens up or down) Value of “a” determines stretch or compression ◦If a > 1, the graph is vertically stretched (narrower than original) ◦If 0 < a < 1, the graph is vertically compressed (wider than original)

7. Sketching a Parabola Find vertex from graphing form and plot that point. You will find 2 points on either side of the vertex by doing the following: ◦Starting at the vertex, go over 1 unit and up/down 1 unit (times “a” if there is a stretch factor) ◦Starting at the vertex, go over 2 units and up/down 4 units (times “a” if there is a stretch factor)

8. Example Problem Find a way to change the equations to make the y = x 2 parabola vertically stretched by a factor of 3, open down, move five units up, and move four units to the left. Write your new equation in graphing form. ANSWER: y = -3(x + 4) 2 + 5 Then, sketch the graph on graph paper.

9. Finding the Distance Between Two Points Either plot the points and draw in the slope triangle OR visualize the distance covered in both the horizontal (x) and vertical (y) directions. Use the base and height of the slope triangle OR the distances covered with the Pythagorean Theorem to find the length of the hypotenuse. This length is the distance between the two points.

10. Writing Equations of Lines Given 2 Points Use either a slope triangle OR rise over run to determine the slope of the line. Use the slope as “m” and one of the points on the line as “x” and “y” to solve for “b”. Once you know “m” and “b”, substitute in those values into the slope-intercept form of an equation to write the equation of your line.

11. Example Problems Find the distance between the given points AND write the equation of the line that contains the given points. (3, 6) and (-2, -4) (1, -3) and (-4, 4) (0, 5) and (-2, -6)

12. Factoring First, factor out any common value with each term (greatest common factor - GCF). Use “box and diamond” method to factor your trinomial (3 terms) into the product of two binomials (2 terms each). ◦Add a 0x term if necessary. Only SOLVE FOR X if the trinomial was originally set equal to 0. Otherwise, you are just rewriting the trinomial in factored form.

13. Examples ANSWERS:

14. Modeling Parabolas Sketch the graph according to the information presented in the problem. Use the given information to determine your vertex. Use the vertex and any other point on the graph to find the value of “a”. Write the graphing form of the equation with your values of “a”, “h”, and “k”.

15. Example In a neighborhood water balloon battle, Benjamin has his home base situated 20 feet behind a 30 foot-high fence. Twenty feet away on the other side of the fence is his enemy’s camp. Benjamin uses a water balloon launcher and shoots his balloons so that they just miss the fence and land in his opponent’s camp. Write an equation that, when graphed, will model the trajectory of the water balloon? ANSWER: y = -3/40(x – 20) 2 + 30

16. Homework 2-69 to 2-75 {SKIP 2-72} (p. 81) Begin to study for Individual Quiz (Tues. 9/29) Reminder – NO CALCULATOR!!

17. Simplifying Radicals Use prime factorization to break the value down into its prime factors. Find a matching group of items equal to the index of the radical and bring that item to the front. Anything left will remain under the radical.

18. Examples ANSWERS: