1. Regents Review #3 Functions f(x) = 2x – 5 y = -2x 2 – 3x + 10 g(x) = |x – 5| y = ¾ x y = (x – 1) 2 Roslyn Middle School Research Honors Integrated Algebra

2. Functions What is a function? A relation in which every x-value(input) is assigned to exactly one y-value (output) Which relation represents a Function? xy 26 37 47 xy 26 27 47 Function Not a Function

3. Functions We can recognize functions using the vertical line test Vertical Line Test: If a graph intersects a vertical line in more than one place, the graph is not a function Which graph represents a function? Function Not a function

4. Functions Functions can be written using function notation “f(x)” means f of x Example: f(x) = 2x – 3 means the same as y = 2x – 3 g(x) = 2x – 3 means the same as y = 2x – 3 h(x) = 2x – 3 means the same as y = 2x – 3

5. Functions In this course, we explored four different Function Families 1)Linear Functions 2)Quadratic Functions 3)Exponential Functions 4)Absolute Value Functions

6. Linear Functions Linear Functions “y = mx +b” The best ways to graph a linear function are… 1)Table of Values 2)Slope-Intercept Method (most effective)

7. Linear Functions Table of Values Method Graph 2x – 4y = 12 y = ½ x – 3 xy -4-5 -2-4 0-3 2-2 4 2x – 4y = 12

8. Linear Functions Before we discuss the Slope-Intercept Method, let’s discuss SLOPE Slope is a ratio: Slope Formula = Positive Slope Negative Slope 0 slope Undefined slope Parallel Lines have the same slope Perpendicular lines have opposite, reciprocal slopes

9. Linear Functions Slope-intercept Method y = mx + b m = slope b = y –intercept (0,b) Graph 6x + 3y = 9 y = -2x + 3 m = b = 3 (0, 3) 6x + 3y = 9

10. Linear Functions Horizontal Lines y = b where b represents the y-intercept y = 4 Vertical Lines x = a where a represents the x-intercept x = 4 y = 4 x = 4

11. Linear Functions Writing the Equation of a Line Write the equation of a line that runs through the points (-3,1) and (0,-1) Find the slope (m) (-3,1) (0,-1) Find the y-intercept (b) y = mx + b Write the equation in “y = mx + b” y = x – 1 m = -2/3 1 = (-2/3)(-3) + b 1 = 2 + b -1 = b b = -1

12. Linear Functions Write the equation of a line that is parallel to y – 2x = 4 and runs through the point (-2,4) Find the slope Parallel lines have the same slope y – 2x = 4 y = 2x + 4 m = 2 Find the y-intercept y = mx + b 4 = 2(-2) + b 4 = -4 + b 8 = b b = 8 Write the equation in “y = mx + b” y = 2x + 8

13. Quadratic Functions Quadratic Functions “y = ax 2 + bx + c” How do we graph quadratic functions? 1)Find the coordinates of the vertex 2)Create a table of values 3)Graph a parabola 4)Label the graph with the quadratic equation

14. Quadratic Functions Graph f(x) = x 2 + 4x – 12 f(x) = x 2 + 4x – 12 means y = x 2 + 4x – 12 Finding the coordinates of the vertex Finding the x-coordinate y = x 2 + 4x – 12 a = 1, b = 4, c = -12 x = x = x = -2 Finding the y-coordinate y = x 2 + 4x – 12 y = (-2) 2 + 4(-2) – 12 y = 4 – 8 – 12 y = -16 Vertex = (-2, -16)

15. Quadratic Functions Vertex (-2,-16) xy 1-7 0-12 -15 -2-16 -3-15 -4-12 -5-7 x-intercept (-6,0) x-intercept (2,0) Axis of Symmetry x = -2 f(x) = x 2 + 4x – 12

16. Quadratic Functions The x-intercepts of the graph of a quadratic function are also known as the “roots”. We can identify the “roots” of a quadratic function by looking at the graph of a parabola and locating the x-intercepts. We can also identify the roots algebraically.

17. Quadratic Functions Finding “roots” algebraically Let’s look at our previous example y = x 2 + 4x – 12 In order to find the “roots” (x-intercepts), set y = to zero y = x 2 + 4x – 12 0 = x 2 + 4x – 12 0 = (x + 6)(x – 2) 0 = x + 6 0 = x – 2 -6 = x 2 = x The roots of the function are (0,-6) and (0,2)

18. Quadratic Functions How does a affect the graph of y = ax 2 + bx + c ? 1.If the coefficient of x 2 gets larger, the parabola becomes narrower 2.If the coefficient of x 2 gets smaller, the parabola becomes wider 3.If the coefficient of x 2 is negative, the parabola opens downward

19. Quadratic Functions How does c affect the graph of y = ax 2 + bx + c ? y = x 2 y = x 2 + 5y = x 2 – 5 “c” represents the y-intercept and moves the parabola up and down.

20. Exponential Functions There are two types of Exponential Functions 1)Exponential Growth y = ab x where b > 1 2)Exponential Decay y = ab x where 0 < b < 1

21. Exponential Functions y = 2 x y = ½ x Plots: xy -2¼ ½ 01 12 24 xy -24 2 01 1½ 2¼

22. Exponential Functions Properties of Exponential Functions What happens to y = 2 x when…. 5 is added multiplied by -1 1)y = 2 x + 52) y = -2 x

23. Exponential Functions Exponential Growth Formulay = a(1 + r) t The cost of maintenance on an automobile increases each year by 8%. If Alberto paid $400 this year for maintenance for his car, what will the cost be (to the nearest dollar) seven years from now? y = a(1 + r) t y = 400(1 +.08) 7 y = 400(1.08) 7 y = 685.5297… The cost will be $686.00

24. Exponential Functions Exponential Decay Formulay = a(1 – r) t A used car was purchased in July 1999 for $12,900. If the car loses 14% of its value each year, what was the value of the car (to the nearest penny) in July 2003? y = a(1 – r) t y = 12,900(1 –.14) 4 y = 12,900(.86) 4 y = 7056.4052… The cost of the car was $7056.41

25. Absolute Value Functions Absolute Value Functions “y = |x|” How do you an input an absolute value function into a graphing calculator? 1)Y = 2)Math arrow over to NUM 3)#1 abs( 4)Input x 5)Graph

26. Absolute Value Functions Properties of Absolute Value Functions What happens to y = |x| when…. 5 is added multiplied by -1 1)y = |x| + 5 2) y = -|x|

27. Absolute Value Functions Properties of Absolute Value Functions What happens to y = |x| when a number other than 1 is multiplied by x? 1.As the coefficient of x gets larger, the graph becomes thinner 2.As the coefficient of x gets smaller, the graph becomes wider

28. Regents Review #3 Now it’s time to study! Using the information from this power point and your review packet, complete the practice problems.