1. Identifying Quadratic Functions

2. The function y = x 2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. y = x 2 is the parent function of a quadratic function. A quadratic function is any function that can be written in the standard form y = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The function y = x 2 can be written as y = 1x 2 + 0x + 0, where a = 1, b = 0, and c = 0. y = x 2 is the parent function for quadratic functions.

3. Last semester, you identified linear functions by finding that a constant change in x corresponded to a constant change in y. The differences between y- values for a constant change in x-values are called first differences.

4. Notice that the quadratic function y = x 2 does not have constant first differences. It has constant second differences. This is true for all quadratic functions.

5. Tell whether the function is quadratic. Explain. Since you are given a table of ordered pairs with a constant change in x-values, see if the second differences are constant. Find the first differences, then find the second differences. The function is not quadratic. The second differences are not constant. xy –2 –2 –1 –1 0 1 2 –1–1 0 –2–2 –9–9 7 +7 +1 +7 +1 – 6 +0 +6

6. Since you are given an equation, use y = ax 2 + bx + c. Tell whether the function is quadratic. Explain. y = 7x + 3 This is not a quadratic function because the value of a is 0.

7. Tell whether the function is quadratic. Explain. This is a quadratic function because it can be written in the form y = ax 2 + bx + c where a = 10, b = 0, and c =9. y – 10x 2 = 9 Try to write the function in the form y = ax 2 + bx + c by solving for y. Add 10x 2 to both sides. + 10x 2 +10x 2 y – 10x 2 = 9 y = 10x 2 + 9

8. xy –2 –2 –1 –1 0 1 2 0 1 1 4 4 Tell whether the function is quadratic. Explain. y + x = 2x 2 yes

9. The graph of a quadratic function is a curve called a parabola. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve. Vertex The point in the parabola through which the axis of symmetry will be drawn Minimum The vertex of a parabola that opens “upward” Maximum The vertex of a parabola that opens “downward”

10. Use a table of values to graph the quadratic function. xy –2 –1 –1 0 1 2 0 4343 1313 1313 4343 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve. What are the domain and range of the graph above? D: all real numbers R: y > 0

11. Axis of Symmetry  A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola.

12. Graph the function: y = -x 2 + 3  Determine the vertex.  Find another point.  Reflect it over the axis of symmetry.

13. Some parabolas open upward and some open downward. When a quadratic function is written in the form y = ax 2 + bx + c, the value of a determines the direction a parabola opens. A parabola opens upward when a > 0. A parabola opens downward when a < 0.

14. Tell whether the graph of the quadratic function opens upward or downward. Explain. Since a > 0, the parabola opens upward. Write the function in the form y = ax 2 + bx + c by solving for y. Add to both sides. Identify the value of a.

15. Example  An acorn drops from a tree branch 20 feet above the ground. The function h = -16t 2 + 20 gives the height h of the acorn (in feet) after t seconds. About how long does it take the acorn to hit the ground? About 1.1 seconds

16. 1. Is y = –x – 1 quadratic? Explain. 2. Graph y = 1.5x 2. No; there is no x 2 -term, so a = 0. Yes, there is an x 2 term Try these…

17. Try these (cont.)… Use the graph for Problems 3-5. 3. Identify the vertex. 4. Does the function have a minimum or maximum? What is it? 5. Find the domain and range. D: all real numbers; R: y ≤ –4 max; – 4 (5, –4)