Translating and the Quadratic Family

1. Translating and the Quadratic Family Lesson 4.4

2. In the previous lesson, you looked at translations of the graphs of linear functions. • Translations can occur in other settings as well. • For instance, what will this histogram look like if the teacher decides to add five points to each of the scores? • What translation will map the black triangle on the left onto its red image on the right?

3. Translations are also a natural feature of the real world, including the world of art. Music can be transposed from one key to another. Melodies are often translated by a certain interval within a composition.

4. In mathematics, a change in the size or position of a figure or graph is called a transformation. • Translations are one type of transformation. • Other types of transformations are reflections, dilations, stretches, shrinks, and rotations. • In this lesson you will experiment with translations of the graph of the function y=x2. • The special shape of this graph is called a parabola. Parabolas always have a line of symmetry that passes through the parabola’s vertex.

6. The parent function is y=x2. • By transforming the graph of a parent function, you can create infinitely many new functions, or a family of functions. • The function y=x2 and all functions created from transformations of its graph are called quadratic functions, because the highest power of x is x2.

7. You will use quadratic functions to • model the height of a projectile as a function of time, or • the area of a square as a function of the length of its side. • The focus of this lesson is on • writing the quadratic equation of a parabola after a translation and • graphing a parabola given its equation. • You will see that success with understanding parabolas will be through studying the location of the vertex.

8. Make My Graph Procedural Note • Different calculators have different resolutions. A good graphing window will help you make use of the resolution to better identify points. • Enter the parent function y=x2 as the first equation. • Enter the equation for the transformation as the second equation. Graph both equations to check your work.

9. Each graph below shows the graph of the parent function y=x2 in black. Find a quadratic equation that produces the congruent, red parabola. Apply what you learned about translations of the graphs of functions in the previous lesson.

10. Write a few sentences describing any connections you discovered between the graphs of the translated parabolas, the equation for the translated parabola, and the equation of the parent function y=x2. • In general, what is the equation of the parabola formed when the graph of y=x2 is translated horizontally h units and vertically k units?

11. Example • This graph shows a portion of a parabola. It represents a diver’s position (horizontal and vertical distance) from the edge of a pool as he dives from a 5 ft long board 25 ft above the water. • Identify points on the graph that represent • when the diver leaves the board, • when he reaches his maximum height, and • when he enters the water. (7.5,30) (5,25) (13.6,0)

12. Example • This graph shows a portion of a parabola. It represents a diver’s position (horizontal and vertical distance) from the edge of a pool as he dives from a 5 ft long board 25 ft above the water. • Sketch a graph of the diver’s position if he dives from a 10 ft long board 10 ft above the water. (Assume that he leaves the board at the same angle and with the same force.)

13. Example • This graph shows a portion of a parabola. It represents a diver’s position (horizontal and vertical distance) from the edge of a pool as he dives from a 5 ft long board 25 ft above the water. • In the scenario described in part b, what is the diver’s position when he reaches his maximum height? (7.5,30) translates to (7.5+5, 30-15) or (12.5, 15)