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 Quadratic Equation – Equation in the form y=ax 2 + bx + c.  Parabola – The general shape of a quadratic equation. It is in the form of a “U” which may open upward or downward.  Vertex – The maximum or minimum point of a parabola.  Maximum – The highest point (vertex) of a parabola when it opens downward.  Minimum – The lowest point (vertex) of a parabola when it opens upward.  Axis of symmetry – The line passing through the vertex having the equation about which the parabola is symmetric.

How does the sign of the coefficient of x 2 affect the graph of a parabola? On your graphing calculator, do the following: 1. Press the Y= key. 2. Clear any existing equations by placing the cursor immediately after the = and pressing CLEAR. 3. Enter 2x 2 after the Y 1 = by doing the following keystrokes. 2 X,T,  x 2 4. Press GRAPH.

Repeat using the equation y = -2x 2. When the coefficient of x 2 is positive, the graph opens upward. When the coefficient of x 2 is negative, the graph opens downward.

How does the value of a in the equation ax 2 + bx + c affect the graph of the parabola?  Clear the equations in the Y= screen of your calculator.  Enter the equation x 2 for Y 1.  Enter the equation 3 x 2 for Y 2. Choose a different type of line for Y 2 so that you can tell the difference between them.  Press GRAPH.

 Clear the second equation in the Y= screen and now enter the equation y = (1/4)x 2.  Press the GRAPH key and compare the two graphs.

Summary for ax 2 When a is positive, the parabola opens upward. When a is negative, the parabola opens downward. When a is larger than 1, the graph will be narrower than the graph of x2.x2. When a is less than 1, the graph will be wider (broader) than the graph of x2.x2.

How does the value of c affect the graph of a parabola when the equation is in the form ax 2 + c? o In the Y= screen of the graphing calculator, enter x 2 for Y 1. o Enter x 2 + 3 for Y 2. o Press the GRAPH key.

Now predict what the graph of y = x 2 – 5 will look like.  Enter x 2 for Y 1 in the Y= screen.  Enter x 2 – 5 for Y 2  Press GRAPH.

What happens to the graph of a parabola when the equation is in the form (x-h) 2 or (x+h) 2 ?  Enter x 2 for Y 1 in the Y= screen.  Enter (x-3) 2 for Y 2.  Press GRAPH.

 Clear the equation for Y 2.  Enter (x+4) 2 for Y 2.  Press GRAPH.

 The vertex of the graph of ax 2 will be at the origin.  The vertex of the graph of the parabola having the equation ax 2 + c will move up on the y-axis by the amount c if c>0.  The vertex of the graph of the parabola having the equation ax 2 + c will move down on the y-axis by the absolute value of c if c<0.  The vertex of the graph of the parabola in the form (x-h) 2 will shift to the right by h units on the x-axis.  The vertex of the graph of the parabola in the form (x+h) 2 will shift to the left by h units on the x-axis.

Compare the graphs of the following quadratic equations to each other. Check your work with your graphing calculator. 1) x 2, x 2 – 7, (x +2) 2 2) 2x 2, x 2 + 6, (1/3)(x-5) 2

Problem 1  All three graphs have the same shape.  The vertex of the graph of x 2 – 7 will move down 7 on the y-axis.  the vertex of the graph of (x+2) 2 will move left two on the x-axis.

Problem 2  The graph of 2x 2 will be the narrowest. The graph of (1/3)(x-2) 2 will be the broadest.  The vertex of x 2 + 6 will be shifted up 6 units on the y- axis compared to the graph of 2x 2.  The vertex of (1/3)(x-2) 2 will be shifted right two units on the x-axis compared to the graph of 2x 2.

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