Here are two equations. One defines function and the other defines function .

1. Show that the expressions defining and are equivalent.
2. What is the vertex of the graph of ? Explain how you know.
3. What are the -intercepts of the graph of ? Explain how you know.

For each equation, write the coordinates of the vertex of the graph that represents the equation.

For each function, write the coordinates of the vertex of its graph, and tell whether the graph opens upward or downward.

Here is a graph that represents .

1. Describe what would happen to the graph if the original equation were modified as follows:

   

   A curve in an x y plane, origin O. Horizontal axis, scale negative 8  to 8, by 2’s. Vertical axis, scale negative 12 to 12, by 2’s. A curve, labeled y equals x squared, passes through the points negative 2 comma 4, 0 comma 0, and 2 comma 4.

2. Sketch the graph of the equation on the same coordinate plane as .

Here are four graphs. Match each graph with a quadratic equation that it represents.

A curve in an x y plane, origin O. Horizontal axis, scale negative 6  to 4, by 1’s. Vertical axis, scale negative 4 to 4, by 1’s. A curve passes through the points 1 comma 0, 2 comma negative 1, 3 comma 0.

A curve in an x y plane, origin O. Horizontal axis, scale negative 6  to 4, by 1’s. Vertical axis, scale negative 4 to 4, by 1’s. A curve passes through the points negative 1 comma 0, negative 2 comma negative 1, negative 1 comma 0.

A curve in an x y plane, origin O. Horizontal axis, scale negative 6  to 4, by 1’s. Vertical axis, scale negative 4 to 4, by 1’s. A curve passes through the points negative 1 comma 2, 3 comma negative 0, 1 comma 2.

A curve in an x y plane, origin O. Horizontal axis, scale negative 6  to 4, by 1’s. Vertical axis, scale negative 4 to 4, by 1’s. A curve passes through the points negative 1 comma negative 2, 0 comma negative 3, and 1 comma negative 2.