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1.6: Relations Agenda 1) Check HW/Warm-Up 2) 1.6 Notes 3) Exit Slip Target: Represent and interpret graphs of relations Warm-up Pg. 1 1. A car is currently 300 miles from its destination and is traveling against the wind. The car travels 60 miles per hour (mph) when there is no wind. The carβs distance from its destination is given by the formula π· = 300 β β(60 β π). Given: D = distance in miles, h = number of hours, c = speed of the wind in mph. What is a correct formula for the carβs distance from its destination after 4 hours? A) π· = 17400 β 290π C) π· = 60 β π B) π· = 17400 β π D) π· = 60 + 4π Warm-up Pg. 2 Solve each equation. 2. 6 β 42 7 4. If 8 = +π¦ =4 112 , π₯ 3. 3 + 42 β 9 π = 90 then what is 3π₯? Coordinate System: The grid formed by the intersection of two number lines, the horizontal axis and the vertical axis. Ordered pair: a set of numbers or coordinates used to locate any point on a coordinate plane, written in the form (x, y). x-coordinate: the first number in an ordered pair. y-coordinate: the second number in an ordered pair. Relation: a set of ordered pairs Mapping: illustrates how each element of the domain is paired with an element in the range. Domain: the set of first numbers of the ordered pairs in a relation. Range: the set of second numbers of the ordered pairs in a relation. In the relation above, the domain is {-2, 1, 0} and the range is {-3, 2, 4} Example 1 a) Express {(4, 3), (-2, -1), (2, -4), (0, -4)} as a table, a graph, and a mapping. b) Determine the domain and range of the relation. Independent variable: the variable in a relation with a value that is subject to choice. Dependent variable: the variable in a relation with a value that depends on the value of the independent variable. Example 3 Identify the independent and the dependent variable for each relation. a) In warm climates, the average amount of electricity used rises as the daily average temperature increases and falls as the daily average temperature decreases. b) The number of calories you burn increases as the number of minutes that you walk increases. Example 3 A relation can be graphed without a scale on either axis. These graphs can be interpreted by analyzing their shape. Describe what is happening in each graph. a) b) Homework <6> p.43 #12, 16, 18, 20, 34, 36 Donβt let this happenβ¦ Brain Break Red in the Face Ice Cube Blanket Section 1.7 Warmup 1.) Draw a graph that represents you taking a dog for a walk. Let your labels be distance and time. You stopped twice to talk to two different neighbors and you ran the last half block to your house. 2.) 1. Express the relation β1, 0 , 2, β4 , β3, 1 , 4, β3 as a table, a graph, and a mapping. Then determine the domain and range. Warmup Problem 2: Algebra 1 Unit 1 Section 1.7 Notes: Functions Function: a relationship between input and output. In a function there is exactly one output for each input. Example 1: Determine whether each relation is a function. Explain. a) b) Example 1: Determine whether each relation is a function. Explain. c) Discrete function: a function of points that are not connected. Continuous function: a function that can be graphed with a line or a smooth curve. Example 2 Circle the continuous functions below. B A D C E Example 3: There are three lunch periods at a school. During the first period, 352 students eat. During the second period 304 students eat. During the third period, 391 students eat. a) Make a table showing the number of students for each of the three lunch periods. b) Determine the domain and range of the function. Example 3 Continued: c) Write the data set of ordered pairs then graph the data. d) State whether the function is discrete or continuous. Explain your reasoning. Vertical line test: if any vertical line passes through no more than one point of the graph of a relation, then the relation is a function. *Use when you are given a graph. Example 4: Determine whether the following graphs represent functions. a) b) c) A function can be represented in different ways. Function Notation: A way to name a function that is defined by an equation. In function notation, the equation π¦ = 3π₯ β 8 is written π π₯ = 3π₯ β 8. It is said βf of xβ. If a number is inside the parenthesis than that is the number you substituted in for x. Example 5: For π π₯ = 3π₯ β 4, find each value. a) π(4) b) Find the missing values in the table using the function above. Nonlinear function: a function with a graph that is not a straight line. Example 6: If β π‘ = 1248 β 160π‘ + 16π‘ 2 , find each value. a) β(3) b) β(2π§) Brain Break A Walk in the Park Down to Earth All Mixed Up Algebra 1 Unit 1 Section 1.8 Notes: Interpreting Graphs of Functions y-intercept: the y β coordinate of a point where a graph crosses the y β axis. x-intercept: the x β coordinate of a point where a graph crosses the x β axis . Example 1: The graph shows the cost at a community college y as a function of the number of credit hours taken x. a) Identify the function as linear or nonlinear. b) Estimate and interpret the intercepts of the function. c) Approximate the cost of a student taking 4 credit hours. Line Symmetry: if a vertical line is drawn and each half of the graph on either side of the line matches exactly. Example 2: The graph shows the cost y to manufacture x units of product. Describe and interpret any symmetry. Brain Break Q: What word becomes shorter when you add two letters to it? A: Short Q: What can you catch but not throw? A: A cold. Q: Forward I am heavy, backward I am not. What am I? A: A ton Example 3: The graph shows the population y of deer x years after the animals are introduced on an island. Estimate when the deer population is: A) positive B) negative C) increasing D) decreasing E) at its maximum population (relative maximum) F) at its minimum population (relative minimum) Example 3: The graph shows the population y of deer x years after the animals are introduced on an island. β’ Describe the end behavior of the graph. β’ How many deer will there be after 0.5 years? 5 years? β’ The deer population started at about 80 deer. When will the deer population be at 80 again? Exit Slip 1. Where do you find intercepts? 2. Using the graph below, estimate what will the companies maximum revenue be. Homework p.52 #20, 22, 24, 26, 30, 34, 46, 48 p.59 #4, 6, 8, 12, 14, 16, 18 Donβt let this happenβ¦