IM Alg1.6.17 Lesson: Changing the Vertex

Here are graphs representing the functions  and , given by  and . Which graph represents each function? Explain how you know.

Where does the graph of  meet the -axis? Explain how you know.

How would you change the equation  so that the vertex of the graph of the new equation is located at the following coordinates and the graph opens as described? , opens up

, opens up

, opens down

Kiran graphed the equation  and noticed that the vertex is at . He changed the equation to  and saw that the graph shifted 3 units to the right and the vertex is now at . Next, he graphed the equation , observed that the vertex is at . Kiran thought, “If I change the squared term  to , the graph will move 5 units to the right and the vertex will be at .” Do you agree with Kiran? Explain or show your reasoning.

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall. Mai uses the equation  to represent the path of the jump.  represents the height of the peanut as a function of the horizontal distance it travels . On the screen, the base of the wall is located at , with the top of the wall at . The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall. What are the values of  and  in this equation?

The 3 arcs on the graph all represent quadratic functions that were initially defined by , but whose equations were later modified. Write equations to represent each curve in the smiley face.

What domain is used for each function to create this graph?