Can we get information about the graph from the equation? Do Now: Graph the equation y=x2

What happens if we change the value of a? When we change the value of a, the shape of the graph will change. If a is positive, then the parabola will open up. If a is negative, the parabola will open down. If a is greater than one, the graph will be narrower. If a is less than one, the graph will be wider.

Y=x2, y=2x2, y=-x2, y=.5x2

How else can the graph change? The graph can also shift up and down. If a constant is added or subtracted to the standard y=x2 equation, then the graph will shift up or down an equal number of units.

y=x2+3, y=x2-3

Can the graph shift side to side? Yes, this happens when we add or subtract a constant to x BEFORE we square it. Instead of the standard y=x2 equation, the equation looks like y=(x-2)2 or y=(x+2)2 If the number is positive, the shift is to the right, if it is negative, the shift is to the left.

y=(x-2)2, y=(x+2)2

How can we get this information if we have an equation in standard form? You already have the tools needed to pick out these shifts just from the standard form. Complete the square Do all the same steps as before, but instead of taking the square root of both sides, solve for x

Example Compare the graph of y=x2-2x-3 to the standard y=x2 Y=x2-2x-3 The graph shifts down 4 and to the right 1 Original equation Get x’s alone Add half of b squared Factor Isolate y Explain differences

Terminology When we have an equation written in the form y=a(x-h)2+k, we know the vertex of the parabola. The vertex is the maximum or minimum of the graph and is located at (h, k) Note: The x value of the vertex can be found by finding -b/2a from the standard equation. Then plug this value into the equation to find the y value. The line x=h is called the axis of symmetry

Summary How can completing the square give us information about the graph of a quadratic equation? HW: pg 95, #1, 3