Quadratic Equation Equation in the form yax 2

ØQuadratic Equation – Equation in the form y=ax 2 + bx + c. ØParabola – The general shape of a quadratic equation. It is in the form of a “U” which may open upward or downward. ØVertex – The maximum or minimum point of a parabola. ØMaximum – The highest point (vertex) of a parabola when it opens downward. ØMinimum – The lowest point (vertex) of a parabola when it opens upward. ØAxis 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 2 x 2 after the Y 1= by doing the following keystrokes. 2 4. X, T, Press GRAPH. x 2

Repeat using the equation y = -2 x 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? v. Clear the equations in the Y= screen of your calculator. v. Enter the equation x 2 for Y 1. v. 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. v. Press GRAPH.

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

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

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. v Enter x 2 for Y 1 in the Y= screen. v Enter x 2 – 5 for Y 2 v Press GRAPH.

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

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

v The vertex of the graph of ax 2 will be at the origin. v 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. v 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. v 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. v 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) 2 x 2, x 2 + 6, (1/3)(x-5)2

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

Problem 2 § The graph of 2 x 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 yaxis compared to the graph of 2 x 2. § The vertex of (1/3)(x-2)2 will be shifted right two units on the x-axis compared to the graph of 2 x 2.