Preparing for Success in Algebra Demonstration Center

1. Preparing for Success in AlgebraDemonstration Center A Collaboration among: Los Angeles USD University of California, San Diego San Diego State University University of California, Irvine

3. From The Draft Ca CC Framework for Algebra 1- April 2013 “The main purpose of Algebra I is to develop students’ fluency with linear, quadratic and exponential functions. The critical areas of instruction involve deepening and extending students’ understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend.”

4. Algebra1Draft framework (continued) “In addition, students engage in methods for analyzing, solving, and using exponential (and) quadratic functions. Some of the overarching ideas in the Algebra I course include: the notion of function, solving equations, rates of change and growth patterns, graphs as representations of functions, and modeling.” Exp Worksheet

5. Worksheet Answer Key 1) Quadratic 2) Linear 3) Exponential 4) Exponential 5) Linear 6) Quadratic 7) Quadratic 8) Linear 9) Linear 10) Exponential

6. Interpreting Functions Alg 1CC • Understand the concept of a function and use function notation. • Interpret functions that arise in applications in terms of the context. • Analyze functions using different representations

7. Building Functions • Build a function that models a relationship between two quantities. • Build new functions from existing functions.

8. Linear, Quadratic, and Exponential Models • Construct and compare linear, quadratic, and exponential models and solve problems. • Interpret expressions for functions in terms of the situation they model.

12. f has Domain R and Co-domain Z f R Z Co-domain Domain f(x) y x Pre-image of y Image of x

13. Range vs Co-domain The range of a function f is the subset of the co-domain of all elements that have a pre-image i.e. the set of all elements of the form f(x) for some x in the domain of f. This is the ‘usual’ definition but use care as some authors use range instead of co-domain and image for what we refer to as range.

14. a e i o 1 2 3 4 5 a e i o 1 2 3 4 5 A one-to-one function A function that is not one-to-one One-to-one functions A function is one-to-one if each element in the range has a unique pre-image

15. a e i o u 1 2 3 4 a e i o 1 2 3 4 5 An onto function A function that is not onto Onto functions A function is onto if there are no elements in the co-domain that are not in the range.

16. a b c 1 2 3 4 a b c d 1 2 3 a b c d a b c d 1 2 3 4 1 2 3 4 a b c 1 2 3 4 Onto vs. one-to-one Are the following functions onto, one-to-one, both, or neither? 1-to-1, not onto Both 1-to-1 and onto Not a valid function Onto, not 1-to-1 Neither 1-to-1 nor onto

17. Functions may be represented • Verbally - e.g. The perimeter of a rectangle is the sum of the lengths of the sides. • Visually – e.g. seismograph readings or electrocardiograms • Algebraically – e.g. y = 2x+1 • Numerically e.g. - A table of values

18. Method of finite differences Assume we have a set of data with equally spaced x-values. The method of finite differences may be applied to determine if there is a linear or quadratic polynomial that fits the data. The technique analyzes the differences in y-values repeatedly to determine when (or if) they become constant.

19. The linear equation is y = 3x-1 The difference is constant so the points have a linear model y = mx+b First page of Charts Handout

20. y = 3x+2

21. F = 9/5 C+32 Since the difference is constant we can model using the linear equation F = mC+b Page 5 of Charts handout

22. April Draft CDE Document “ Also new to the Algebra I course is standard F-BF.4, in which students find inverse functions in simple cases. For example, an Algebra I student can solve the equation 𝐶 =9/5 𝐹+32 for 𝐹. The student starts with this formula, showing how Celsius temperature is a function of Fahrenheit temperature, and by solving for 𝐹 finds the formula for the inverse function. “

23. CC Algebra 1 (Continued) “This is a contextually appropriate way to find the expression for an inverse function, in contrast with the practice of simply swapping 𝑥 and 𝑦 in an equation and solving for 𝑦. “

24. Finite Differences for Quadratics The power of the finite differences method becomes evident with data that has a quadratic model i.e. a quadratic equation that contains all the data points. Here we want to take differences of differences to see if they became constant.

25. The quadratic equation is y = 2x2+x-3 c = -3 a+b+c = 0 b = 1 a = 2 The second difference is constant so the points have a quadratic model y = ax2+bx+c Page 2 of Chart Handout

27. Finite Differences for Exponentials • Exponential functions are not polynomials of any degree. What happens if you apply the method of successive differences to an exponential? It should mean that the successive differences never become constant Page 3 of the Charts Handout

29. The exponential equation is y = 3 2x The quotient is constant so the points have an exponential model y = f(0)2x/h =f(0)2x Page 4 of Charts Handout

30. y = 2x2+x-1 a=2, c = -1, a+b+c = 2, b = 1

31. Exponential worksheet