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GRAPHS OF QUADRATIC EQUATIONS

GRAPHS OF QUADRATIC EQUATIONS. Axis of symmetry – The line passing through the vertex having the equation about which the parabola is symmetric. Vocabulary. Quadratic Equation – Equation in the form y=ax 2 + bx + c.

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GRAPHS OF QUADRATIC EQUATIONS

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  1. GRAPHS OF QUADRATIC EQUATIONS

  2. Axis of symmetry – The line passing through the vertex having the equation about which the parabola is symmetric. Vocabulary • Quadratic Equation – Equation in the form y=ax2 + 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.

  3. Shapes of Graphs How does the sign of the coefficient of x2 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 2x2 after the Y1= by doing the following keystrokes. 2 X,T, x2 4. Press GRAPH.

  4. Up or Down Repeat using the equation y = -2x2. When the coefficient of x2 is positive, the graph opens upward. When the coefficient of x2 is negative, the graph opens downward.

  5. Narrow or Wide? How does the value of a in the equation ax2 + bx + c affect the graph of the parabola? • Clear the equations in the Y= screen of your calculator. • Enter the equation x2 for Y1. • Enter the equation 3 x2 for Y2. Choose a different type of line for Y2 so that you can tell the difference between them. • Press GRAPH.

  6. More Narrow or Wide • Clear the second equation in the Y= screen and now enter the equation y = (1/4)x2. • Press the GRAPH key and compare the two graphs.

  7. Summary for ax2 • 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. • When a is less than 1, the graph will be wider (broader) than the graph of x2.

  8. Crossing the y-axis How does the value of c affect the graph of a parabola when the equation is in the form ax2 + c? • In the Y= screen of the graphing calculator, enter x2for Y1. • Enter x2 + 3 for Y2. • Press the GRAPH key.

  9. Higher or Lower Now predict what the graph of y = x2 – 5 will look like. • Enter x2for Y1 in the Y= screen. • Enter x2 – 5for Y2 • Press GRAPH.

  10. Left or Right? What happens to the graph of a parabola when the equation is in the form (x-h)2 or (x+h)2? • Enter x2 for Y1 in the Y= screen. • Enter (x-3)2 for Y2. • Press GRAPH.

  11. Which Way? • Clear the equation for Y2. • Enter (x+4)2for Y2. • Press GRAPH.

  12. Vertex Summary • The vertex of the graph of ax2 will be at the origin. • The vertex of the graph of the parabola having the equation ax2 + cwill move up on the y-axis by the amount c if c>0. • The vertex of the graph of the parabola having the equation ax2 + 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)2will shift to the left by h units on the x-axis.

  13. Practice Problems Compare the graphs of the following quadratic equations to each other. Check your work with your graphing calculator. 1) x2, x2 – 7, (x +2)2 2) 2x2, x2 + 6, (1/3)(x-5)2

  14. Problem 1 • All three graphs have the same shape. • The vertex of the graph of x2 – 7 will move down 7 on the y-axis. • the vertex of the graph of (x+2)2will move left two on the x-axis.

  15. Problem 2 • The graph of 2x2will be the narrowest. The graph of (1/3)(x-2)2will be the broadest. • The vertex of x2 + 6 will be shifted up 6 units on the y-axis compared to the graph of 2x2. • The vertex of (1/3)(x-2)2will be shifted right two units on the x-axis compared to the graph of 2x2.

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