1. Introduction to Quadratics Objectives:  Define Quadratic Functions and Parent functions  Explore Parameter changes and their effects. R. Portteus 2005-06 T. Merrill 2006

2. What is a Quadratic Function?  A quadratic function is any function whose graph is a parabola.  A quadratic equation is any equation that can be written in the form y = ax 2 + bx + c. The constants a, b, and c are called the parameters of the equation. These values tell us the shape and location of the parabola.

3. Parent Functions  The parent function is the simplest function of a certain type. It is called this because all of functions in that group look like that and only change location and shape.  The linear parent function is y = x.  All graphs are straight lines.  The quadratic parent function is y = x 2.  All graphs are parabolas.

4. Effects on Parameters  What happens to the graph of y = ax 2 when a is changed?  If a > 0, the parabola opens upward.  If a < 0, the parabola opens downward.

5. Examples y = 2x 2 opens upwardy = -2x 2 opens downward

6. Graph the following four parabolas on the graph at right and label each A.B.C.D. x y Effect of –a In a quadratic formula, when a is multiplied by -1 the resulting graph is the same as the original graph _____________________________. Reflected over the x-axis

7. Effects on Parameters  If two quadratic functions of the form y = ax 2 have different coefficients, then one graph will be wider than the other.  Which of these functions produce the widest graph?  y = 3x 2, y = -5x 2, y = ¾ x 2

8. Answer y = ¾ x 2 y = -5x 2 y = 3x 2 y = ¾ x 2 is wider, so the smaller a is, the wider the graph is.

9. Graph the following four parabolas on the graph at right and label each A.B.C.D. x y Effect of |a| In a quadratic formula, when |a| increases, the resulting graph is ___________, when |a| decreases, the resulting graph is _____________. narrower wider

10. Effects on Parameters  If two quadratic functions of the form y = (x - h) 2 have different values for h then one graph will be a translation right or left from the other graph.  Compare these three graphs: y = (x – 0) 2 y = (x + 1) 2 y = (x – 4) 2 Parent function! What happens to the parent function when we “add 1”?What happens to the parent function when we “add 1”? What happens when we “subtract four”?What happens when we “subtract four”?

11. x yAnswer y = (x+1) 2 y = x 2 - 4 y = (x + 1) 2 is translated left one spaces from y = x 2 y = (x – 4) 2 is translated right four spaces from y = x 2

12. Effects on Parameters  If two quadratic functions of the form y = x 2 + c have different constants, c, then one graph will be a translation up or down from the other graph.  Compare the graphs of y = x 2 + 3 with the graph of y = x 2 – 4.

13. Answer y = x 2 + 3 y = x 2 - 4 y = x 2 + 3 is translated up three spaces from y = x 2 and y = x 2 – 4 is translated down four spaces from y = x 2, so there are 7 spaces in between the two graphs.

14. Graph the following four parabolas on the graph at right and label each A.B.C.D. x y Effect of c In a quadratic formula, when c increases one unit, the resulting graph is __________________, when c decreases one unit, the resulting graph is __________________. translated up 1 unit translated down 1 unit

15. Examples  Identify the parent function of the following equations. 1. y = -3x 2 + 4x – 7? y = x 2 → Quadratic Function 2. 3x – 4y = 8 y = x → Linear Function

16. Examples - Continued Determine whether the function face upward or downward. 1. y = -3x 2 + 2 - Downward 2. y = ½ x 2 - Upward

17. Examples- Continued What is the new equation if the given function is translated down 4 spaces? 1. y = x 2 – 3  y = x 2 – 7 2. y = x 2 + 8  y = x 2 + 4

18. Examples - Continued  Order the equations from widest to most narrow. 1. y = -6x 2, y = ¼ x 2, y = 4x 2 How do the two given equation compare? 2. y = -3x 2 – 8 y = x 2 y = ¼ x 2, y = 4x 2, y = -6x 2 Translated down 8 spaces, reflected over the x-axis to face downwards and is narrower.

19. Lesson Check  What causes a parabola to move left or right?  What causes a parabola to move up or down?  What causes a parabola to flip upside down?  What causes a parabola to get wider?  What causes a parabola to get narrower?