1. 10-1/10-2 Graphing Quadratic Functions By Ms. Hall

2. Do you see a parabola? Can you see a non-function?

3. 10-1 Exploring Quadratic Functions The graph of a quadratic function is a parabola or u-shape. A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. The vertex is a minimum. If the parabola opens down, the vertex is the highest point. The vertex is a maximum. NOTE: if the parabola opened left or right it would not be a function! It fails the vertical line test! Vertex

4. Line of Symmetry Where have you heard this before? y x Line of Symmetry The line of symmetry ALWAYS passes through the vertex. Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side. Vertex

5. Example: Identify the vertex of each graph. Tell whether it is a minimum or a maximum. Can you tell me how you are going to tell the difference? a. The vertex is (1, 2). b. The vertex is (2, –4). It is a maximum. It is a minimum.

6. Let’s make some quadratic graphs! Go to the next page and find the blank space at the bottom of the paper. You should see the heading Examples 1-5. Let’s graph Example 1. ***Be sure and label it.

7. Example 1: Make a table of values and graph the quadratic function y = x 2. You pick values for x then plug them back into the equation to find y. We are going to start by picking 3 values for x but we may have to pick more until we see our parabola. 1313 x y = x 2 (x, y) 1313 0 (0) 2 = 0 (0, 0) 1313 2 (2) 2 = 1 (2, 1 ) 1313 1313 1313 3 (3) 2 = 3 (3, 3) 1313

8. Example 2: Make a table of values and graph the quadratic function y = -2x 2. You pick 3 values for x then plug them back into the equation to find y. Again, you might need to pick more to make sure you see your parabola! x y = -2x 2 (x, y) 0 -2(0) 2 = 0 (0, 0) 1 -2(1) 2 = -2 (1, -2) 2 -2(2) 2 = -8 (2, -8)

9. Various Forms of Quadratic Equations Vertex Form is one such form. What are the other forms…patience, patience! Vertex form is awesome because it allows you to see the vertex which is the MIDDLE of the graph. So you instantly know your vertex as well as your line of symmetry! So how would that help you graph?

10. Vertex Form So what does it have to do with quadratic graphs? Let’s compare two quadratic functions and see if we notice anything.

11. Let’s start by comparing the most basic quadratic equation y = x 2 and others. Example 3: Graph the quadratic functions y = 3x 2 and y = 3x 2 – 2. Compare the graphs. The graph of y = 3x 2 – 2 has the same shape as the graph of y = 3x 2, but it is shifted down 2 units. x y = 3x 2 y = 3x 2 – 2 212 10 13 1 00  2 -1 3 1 -212 10

12. Let’s start by comparing the most basic quadratic equation y = x 2 and others Example 4: Graph the quadratic functions y = 4x 2 and y = 4x 2 + 3. Compare the graphs. The graph of y = 4x 2 + 3 has the same shape as the graph of y = 4x 2, but it is shifted up 3 units. x y = 4x 2 y = 4x 2 + 3 216 19 14 7 00 3 -1 4 7 -216 19

13. Example:  (x) = –x 2,  (x) = –3x 2, and  (x) = x 2 Which one is the widest? Narrowest? So, the order from widest to narrowest is  (x) = x 2,  (x) = –x 2,  (x) = –3x 2. 1212 1212  (x) = –x 2  (x) = x 2 1212 Of the three graphs,  (x) = x 2 is the widest and  (x) = –3x 2 is the narrowest. 1212  (x) = –3x 2

14. a tells you how wide or narrow the parabola is. If |a| < 1, then the parabola is wide. If |a| > 1, then the parabola is narrow. y x a tells you if the parabola opens up or down. If a is positive, then the parabola opens up. If a is negative, then the parabola opens down.

15. What about h and k? y = a(x-h) 2 + k k moves the parabola up and down the y-axis. To move the parabola up, k needs to be positive. To move the parabola down, k needs to be negative. h moves the parabola left and right across the x-axis. To move the parabola to the left, h needs to be positive. To move the parabola to the right, h needs to be negative. y x

16. Height h is dependent on time t. Graph t on the x-axis and h on the y-axis. Use positive values for t. Example 5: A monkey drops an orange from a branch 26 ft above the ground. The force of gravity causes the orange to fall toward Earth. The function h = –16t 2 + 26 gives the height of the orange, h, in feet after t seconds. Graph this quadratic function. th = –16t 2 + 26 026 110 2–38