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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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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

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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.

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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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WARM-UP: Graphing Using a Table x y = 3x 2 y -2 y = 3(-2) 2 -8 y = 3(-1) 2 -5 0 y = 3(0) 2 -2 1 y = 3(1) 2 1 2 y = 3(2) 2 4 GRAPH. y = 3x

WARM-UP: Graphing Using a Table x y = 3x 2 y -2 y = 3(-2) 2 -8 y = 3(-1) 2 -5 0 y = 3(0) 2 -2 1 y = 3(1) 2 1 2 y = 3(2) 2 4 GRAPH. y = 3x

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