1. Essential Question: What is the shape of a quadratic graph?

2. 10-1: Exploring Quadratic Graphs Quadratic Function: a function that can be written in the form y = ax 2 + bx + c. This is called the standard form of a quadratic function. Examples: y = 5x 2 Examples: y = x 2 + 7 Examples: y = x 2 – x – 3 The simplest quadratic function is y = x 2. This is called the quadratic parent function. The graph of a quadratic function is a U-shaped curve called a parabola.

3. 10-1: Exploring Quadratic Graphs You can fold a parabola so that the two sides match exactly. This property is called symmetry. The fold/line that divides the parabola into two equal parts is called the axis of symmetry. The highest or lowest point on a parabola is called the vertex, which is always on the axis of symmetry. If a > 0 (is positive) in ax 2 + bx + c, the parabola opens up, and the vertex represents the minimum of the graph. If a < 0 (is negative) in ax 2 + bx + c, the parabola opens down, and the vertex represents the maximum of the graph.

4. 10-1: Exploring Quadratic Graphs Example 1: Identifying a Vertex Identify the vertex of each graph. Tell whether it is a minimum or maximum The vertex is at (1, -2)The vertex is at (-2,4) It is a minimumIt is a maximum

5. 10-1: Exploring Quadratic Graphs Y OUR T URN Identify the vertex of each graph. Tell whether it is a minimum or maximum The vertex is at (4, 3) It is a maximum The vertex is at (-3, -3) It is a minimum

6. 10-1: Exploring Quadratic Graphs No need to copy You can use the fact that a parabola is symmetric to graph it quickly. First, find the coordinates of the vertex, then a few points on either side of the vertex. Then reflect the points across the axis of symmetry. For functions in the form y = ax 2, the vertex is at the origin

7. 10-1: Exploring Quadratic Graphs Example 2: Graphing y = ax 2 Make a table of values and graph the quadratic function y = ½ x 2 xy = ½ x 2 (x, y) 0½ (0) 2 = 0(0, 0) 2½ (2) 2 = 2(2, 2) 4½ (4) 2 = 8(4, 8)

8. 10-1: Exploring Quadratic Graphs Y OUR T URN Make a table of values and graph the quadratic function y = -2x 2 xy = -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. 10-1: Exploring Quadratic Graphs Assignment Worksheet #10-1 Problems 1, 3, 5, 13, 15 Graph each function (don’t follow directions for #1 – 5)

10. Essential Question: What is the shape of a quadratic graph?

11. 10-1: Exploring Quadratic Graphs The y-axis is also the axis of symmetry for functions in the form y = ax 2 + c, so like yesterday, start your table of values with x = 0. Fundamentally, the value of c translates (shifts) the graph up or down. Example 4: Graph y = 2x 2 + 3 xy = 2x 2 + 3(x, y) 02(0) 2 + 3 = 3(0, 3) 12(1) 2 + 3 = 5(1, 5) 22(2) 2 + 3 = 11(2, 11)

12. 10-1: Exploring Quadratic Graphs Y OUR T URN Make a table of values and graph the quadratic function y = x 2 – 4 xy = x 2 – 4(x, y) 0(0) 2 – 4 = -4(0, -4) 1(1) 2 – 4 = -3(1, -3) 2(2) 2 – 4 = 0(2, 0)

13. 10-1: Exploring Quadratic Graphs You can model the height of an object moving due to gravity using a quadratic function. As an object falls, its speed continues to increase. Ignoring air resistance, you can find the approximate height of a falling object using the function h = -16t 2 + c Example 5 Suppose you see an eagle flying over a canyon. The eagle is 30 ft above sea level when it drops a rock. The function y = -16t 2 + 30 gives the height of the rock after t seconds. Graph this quadratic function.

14. 10-1: Exploring Quadratic Graphs Example 5 Suppose you see an eagle flying over a canyon. The eagle is 30 ft above sea level when it drops a rock. The function h = -16t 2 + 30 gives the height of the rock after t seconds. Graph this quadratic function. th = -16t 2 + 30(t, h) 0-16(0) 2 + 30 = 30(0, 30) 1-16(1) 2 + 30 = 14(1, 14) 2-16(2) 2 + 30 = -34(2, -34)

15. 10-1: Exploring Quadratic Graphs Y OUR T URN Suppose a squirrel is in a tree 24 ft above the ground. She drops an acorn. The function h = -16t 2 + 24 gives the height of the acorn after t seconds. Graph this function. Graph on board th = -16t 2 + 24(t, h) 0-16(0) 2 + 24 = 24(0, 24) 1-16(1) 2 + 24 = 8(1, 8) 2-16(2) 2 + 24 = -40(2, -40)

16. 10-1: Exploring Quadratic Graphs Assignment Worksheet 10-1 (You got it yesterday) Problems 17 – 27, odds