Translating and the Quadratic Family Lesson 4.4

Translations can occur in other settings as well. 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?

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.

Translations are one type of transformation. 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.

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.

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.

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.

Each graph below shows the graph of the parent function y=x2 in black 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.

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?

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)

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.)

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)