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H. Algebra 2 1.1 Notes 1.1 (Day One) Domain, Range, and End Behavior Date: ____________ Learning Target A: I can represent an interval on a number line using an inequality, set notation, and interval notation. Interval: a part of a number line without any breaks Finite Interval: has two endpoints which may or may not be included in the interval Finite Interval: is unbounded on one or both ends We have different notations we can use to represent intervals on a number line. Fill in the table below with the given information: Description of Interval All real numbers from a to b, including a and b All real numbers greater than a Type of Interval Inequality Set Notation Interval Notation finite πβ€π₯β€π {π₯|π β€ π₯ β€ π} [π, π] infinite π₯>π {π₯|π₯ > π} (π, β) All real numbers less than or equal to a Using what you gathered from above, fill out each table for the given number lines: A) Finite Interval Inequality Set Notation Interval Notation B) Finite Interval Inequality Set Notation Interval Notation 1 H. Algebra 2 1.1 Notes Reflect. Consider the interval shown on the number line. 1. Consider the interval shown on the number line. a) Represent the interval using interval notation b) What numbers are included in the interval? 2. What do the intervals [0, 5], [0, 5), πππ (0, 5) have in common? What makes them different? 3. The symbol βͺ represents the union of two sets. What do you think the notation (ββ, 0) βͺ (0, β) represents? Identifying a Functionβs Domain, Range, and End Behavior from its Graph Learning Target B: I can identify a functionβs domain, range, and end behavior from its graph. Given the function π, define the following: Gv Domain of π: the set of ____________ values, x. Range of π: the set of ____________ values, π(π₯). State the domain and range for the graph of the linear function as an inequality, using set notation, and using interval notation. End Behavior: describes what happens to the π(π₯) values as the π₯-values either increase without bound (approach positive infinity) or decrease without bound (approach negative infinity). For example, using the graph from above: As x increases without bound, or as x approaches positive infinity, we see that the π(π₯) values are also increasing and approaching positive infinity. This is written: Describe the end behavior as x decreases without bound, or as x approaches negative infinity. 2 H. Algebra 2 1.1 Notes For each graph, give the domain and range as an inequality, using set notation, and using interval notation. Then describe the end behavior. A) The graph of the quadratic function , π(π₯) = π₯ 2 is shown. B)The graph of the exponential function π(π₯) = 2π₯ is shown. C) The graph of the absolute value function π(π₯) = |π₯| + 3 is shown. D) The graph of a function is shown. E) π(π₯) = βπ₯ 2 3 H. Algebra 2 1.1 Notes 1.1 (Day Two) Domain, Range, and End Behavior Date: ____________ Review. Graphing Linear Functions Given each linear function, identify its slope and y-intercept, then draw its graph. A) π(π₯) = π₯ + 1 1 B) π(π₯) = 2 π₯ β 3 3 C) π(π₯) = β 2 π₯ + 5 Graphing a Linear Function on a Restricted Domain Learning Target C: I can graph a linear function on a restricted domain. Unless otherwise stated, a function is assumed to have a domain consisting of all real numbers for which the function is defined. Many functionsβsuch as linear, quadratic and exponential functionsβare defined for all real numbers, so their domain is all real numbers, which can be written many different ways. As the real number set: ____________________ As an Inequality: ___________________________ Set Notation: ________________________________ Interval Notation: __________________________ There are some real-world situations that can be modeled by mathematical functions, but the set of all real numbers would not make sense for the situation. In these cases, we need to restrict the domain. If the rule of a function and its restricted domain are given, you can draw the graph of the function and then identify the range. Example 1. For the given function and domain, draw the graph and identify the range using the same notation as the given domain. 3 A) π(π₯) = 4 π₯ + 2 with domain [β4, 4]. 4 H. Algebra 2 1.1 Notes B) π(π₯) = βπ₯ β 2 with domain {π₯|π₯ > β3} Reflect. 1. In Part A, how does the graph change if the domain is (-4, 4) instead of [-4, 4]? 2. In Part B, what is the end behavior as x increases without bound? Why canβt you talk about the end behavior as x decreases without bound? 1 C) π(π₯) = β 2 π₯ + 2 with domain β 6 β€ π₯ < 2 2 D) π(π₯) = 3 π₯ β 1with domain (ββ, 3] 5 H. Algebra 2 1.1 Notes Modeling with a Linear Function Learning Target D: I can model with a linear function. Recall that when a real-world situation involves a constant rate of change, a linear function is a reasonable model for the situation. The situation may require restricting the functionβs domain. Example 2. Write a function that models the given situation. Determine a domain from the situation, graph the function using that domain, and identify the range. A) Joyce jogs at a rate of 1 mile every 10 minutes for a total of 40 minutes. (Use inequalities for the domain and range) Step 1. Identify and define the independent and dependent variables. Step 3. Find the y-intercept. Distance (mi) Step 2. Find the rate of change. Step 4. Write the function. Identify the domain. 5 4 3 2 1 10 Step 5. Graph the function. Identify the range. 20 30 40 50 Time (min) B) A candle 6 inches high burns at a rate of 1 inch every 2 hours for 5 hours. (Use interval notation for domain and range. C) While standing on a moving walkway at an aiport, you are carried forward 25 feet every 15 seconds for one minute. (Use set notation for the domain and range) 6