GRAPHS OF QUADRATIC EQUATIONS SPI: 3102.3.11 Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation.

What is your GOAL and Purpose? Today you are going to graph simple quadratic functions. You already know how to graph linear and exponential functions. This will help you with graphing quadratics. You need to know how to graph simple quadratic functions so you solve a problem involving solar energy as in Example 6 p. 631.

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. Axis of symmetry – The line passing through the vertex having the equation about which the parabola is symmetric.

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.

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.

Stretch or Shrink? 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.

More Stretch or Shrink 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.

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.

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 x2 for Y1. Enter x2 + 3 for Y2. Press the GRAPH key.

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

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.

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

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 + 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 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)2 will shift to the left by h units on the x-axis.

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

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)2 will move left two on the x-axis.

Problem 2 The graph of 2x2 will be the narrowest. The graph of (1/3)(x-2)2 will 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)2 will be shifted right two units on the x-axis compared to the graph of 2x2.

Reflection… Explain how you can tell if the graph of a quadratic function opens up or down.

Extended Writing… Compare and Contrast vertical stretch and vertical shrink.