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Understanding Relations and Functions Through Graphs and Tables

Explore relations and functions through ordered pairs, coordinate planes, domains, ranges, and mapping diagrams. Learn how to determine if a relation is a function using the vertical line test and function rules.

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Understanding Relations and Functions Through Graphs and Tables

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  1. Bellringer Graph each ordered pair on the coordinate plane. 1. (–4, –8) 2. (3, 6) 3. (0, 0) 4. (–1, 3) Evaluate each expression for x = –1, 0, 2, and 5. 6. x + 2 7. –2x + 3 8. 2x2 + 1 9. |x – 3|

  2. Solutions 6.x + 2 for x = –1, 0, 2, and 5: 1, 2, 4, 7 7. –2x + 3 for x = –1, 0, 2, and 5: 5, 3, –1, –7 8. 2x2 + 1 for x = –1, 0, 2, and 5: 3, 1, 9, 51 9. |x – 3| for x = –1, 0, 2, and 5: 4, 3, 1, 2 . . .

  3. Chapter 2.1 Relations and Functions Objective: Students will use tables and graphs to plot points and describe relations and functions

  4. Relations • Relation – a set of pairs of input and output values • Can be written in ordered pairs (x,y) • Can be graphed on a coordinate plane • Domain – the set of all input values • The x values of the ordered pairs • Range – the set of all output values • The y values of the ordered pairs • When writing domains and ranges: • Use braces { } • Do not repeat values NON-Example: {3,3,5,7,9}

  5. Graph the relation {(–3, 3), (2, 2), (–2, –2), (0, 4), (1, –2)}.

  6. {(0,4),(-2,3),(-1,3),(-2,2),(1,-3)} {(-2,1),(-1,0),(0,1),(1,2)} Try These Problems

  7. Write the ordered pairs for the relation. Find the domain and range. {(–4, 4), (–3, –2), (–2, 4), (2, –4), (3, 2)} The domain is {–4, –3, –2, 2, 3}. The range is {–4, –2, 2, 4}.

  8. Mapping Diagram • Another way to represent a relation (beside traditional graphing) • Links elements of the domain with corresponding elements of the range • How To make a mapping diagram: • Make two lists – place numbers from least to greatest • Domains on the left • Ranges on the Right • Draw arrows from corresponding domains to ranges (x’s to y’s)

  9. Make a mapping diagram for the relation {(–1, 7), (1, 3),(1, 7), (–1, 3)}. Domain Range -1 1 3 7

  10. {(0,2),(1,3),(2,4)} {(2,8),(-1,5),(0,8),(-1,3),(-2,3)} Try These ProblemsMake a Mapping Diagram for each relation. -2 -1 0 2 0 1 2 2 3 4 3 5 8

  11. Functions • Function – a relation in which each element of the domain is paired with EXACTLY one element of the range • All functions are relations, but not all relations are functions!!!

  12. There are several ways to determine if a relation is a function: • Mapping Diagram • If any element of the domain (left) has more than one arrow from it • List of ordered pairs • Look to see if any x values are repeated • Coordinate Plane • Vertical Line Test – If a vertical line passes through more than one point on the graph then the relation is NOT a function.

  13. Determine whether each relation is a function. a) b) -1 0 2 3 -2 0 5 -1 3 4 -1 3 5 This is NOT a function because -2 is paired with both -1 and 3. This is a function because every element of the domain is paired with exactly one element of the range.

  14. 2 3 4 7 5 6 8 Try These ProblemsDetermine whether each relation is a function. a) b) -1 0 1 -3 7 10 Not a function Function

  15. Vertical Line Test

  16. Try These ProblemsUse the Vertical Line Test to determine whether each graph represents a function. a) b) c) Not a Function Function Not a Function

  17. Function Rules • Function Rule – expresses an output value in terms of an input value • Examples: • y = 2x • f(x) = x + 5 • C = πd • Function Notation – • f(x) is read as “f of x” • This does NOT mean f times x !!!! • f(3) is read as “f of 3”: It means evaluate the function when x = 3. (plug 3 into the equation) • Any letters may be used C(d) h(t)

  18. 9 1 – 2 9 1 – x 9 –1 ƒ(2) = = = –9 Find ƒ(2) for each function. a.ƒ(x) = –x2 + 1 ƒ(2) = –22 + 1 = –4 + 1 = –3 b.ƒ(x) = |3x| ƒ(2) = |3 • 2| = |6| = 6 c.ƒ(x) =

  19. Try These ProblemsFind f(-3), f(0), and f(5) for each function • f(x) = 3x – 5 f(-3) = 3(-3) – 5 = -9 – 5 = -14f(0) = 3(0) – 5 = 0 – 5 = -5f(5) = 3(5) – 5 = 15 – 5 = 10 • f(a) = ¾ a – 1 f(-3) = ¾ (-3) – 1= -9/4 – 4/4 = -13/4 f(0) = ¾ (0) – 1 = 0 – 1 = -1 f(5) = ¾ (5) – 1 = 15/4 – 4/4 = 11/4 • f(y) = -1/5 y + 3/5 f(-3) = -1/5 (-3) + 3/5 =3/5 + 3/5 = 6/5f(0) = -1/5 (0) + 3/5= 0 + 3/5 = 3/5f(5) = -1/5 (5) + 3/5 = -5/5 + 3/5 = -2/5

  20. Groupwork/Homework • Day 1: • 2.1, page 59 (1-19) • Show all work for function notation problems • Day 2: • Page 59 (22 – 38 even) • Be sure to write each problem – this includes writing out the sets of coordinate pairs and sketching graphs • Show all work for function notation problems

  21. Groupwork/Homework Honors • Section 2.1 page 59 (1 - 37 odd, 55 - 57)

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