4.4 Absolute Value Equations. What is the absolute value of a number? Example 1 Simplify a) 6 b) 4 c) 7 3. Example 2 Solve x = 2

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1 4.4 Absolute Value Equations What is the absolute value of a number? Eample Simplif a) 6 b) 4 c) 7 3 Eample Solve = Steps for solving an absolute value equation: ) Get the absolute value b itself on one side (everthing not in the absolute value should be on the other side). ) Set up two cases: the positive case and the negative case. Solve for each case. 3) Check each solution to see if it is an actual or etraneous solution. Eample 3 Solve + 7 = 0First, b inspection: Positive Case: Negative Case: Check: Eample 4 Solve + 3 = 9 Positive Case: Negative Case: Check:

2 Eample 5 Solve 3 = algebraicall Check: Eample 6 Solve = 4 Check: no solutions An Absolute Value Equation with No Solution: Eample 7 Solve =

3 Eample 8 Solve + 5 = 4 algebraiciall Check: quadratic absolute value equations Eample 9 Solve 7 + = 0 Check: Eample 0 Solve =

4 4.3A Absolute Value Functions Part Eample Graph = The graph = consists of two graphs: This is wh an absolute value graph is called a piecewise function. In general, for absolute value functions: = f() is Graphing Absolute Value Equations To graph = a b + c + d : ) Find the -coordinate of the verte b solving b + c = 0. The verte is ( c b, d). ) Construct a table of values, using values to the left & right of the -coordinate of the verte. 3) Plot the points. The graph is smmetrical about the verte, and opens up if a > 0, down if a < 0. Eample Graph = State the domain, range, intercepts, and write as a piecewise function. Step :

5 Domain, Range, intercepts, piecewise function: Eample 3 Graph = 4. State the domain, range, intercepts, and write as a piecewise function.

6 4.3B Absolute Value Functions Part At times, absolute value functions are set up analogous to quadratic functions in standard form, using h, k, and a values to graph the function. Look back at last da s notes at the graph of =. What is the basic count? Summarize how each component of the function affects the graph: = ±a h + k Eample Graph each function a) = 3 b) = + 4 c) = d) = + +

7 e) = 3 7 f) = 3 4 Eample For e) and f), state the domain, range, intercepts, & write as piecewise functions: If a graph is constructed for an function f(), what will the graph for f() look like? Eample 3 Graph = 4 using a table of values. Then graph = 4 f() f()

8 Eample 4 Given each graph of = f(), graph = f().

9 4.5A Rational Functions Part A function f is a rational function if f() = g(), where g() and h() are polnomials. h() The domain of f consists of all real numbers ecept values that make the denominator equal to zero (undefined values). Eample What are the undefined values for each rational function? a) +5 b) 4 c) +4 5 d) 7 +6 Graphing Rational Functions Eample Graph = The function is not defined for = 0, so this is a vertical asmptote of the function, and is drawn as a dashed line. The pattern in the table can be written as: as 0 +, f() + as 0, f() as +, f() 0 + as, f() 0 The line defined b = 0 is said to be a horizontal asmptote of the function, and is drawn as a dashed line. An asmptote is not part of the graph. A vertical asmptote is a line the graph approaches as the denominator approaches zero. A horizontal asmptote Is a line the graph approaches as gets larger. Not ever rational function has both a horizontal AND vertical asmptote. Vertical Asmptotes are found when the denominator equals zero. Sometimes the denominator must be factored to find the vertical asmptotes (see Eample above).

10 Eample 3 Graph = What is the vertical asmptote? Plot it. Now, complete the table of values below: Another wa to think about the function above is using = h + k, where the intersection of the two asmptotes is (h, k). So, = is just like =, but shifted one unit right and one unit down. Before we graphed = above, we knew the vertical asmptote, but the horizontal asmptote wasn t apparent until after. How can we find the horizontal asmptote before? The horizontal asmptote is the value that approaches as approaches ±. Consider the function f() = g() : h() ) If h() is a higher power than g(), the horizontal asmptote is = k. ) If g() and h() have the same power, the horizontal asmptote is leading coefficient of numerator = leading coefficient of denominator 3) If g() is a higher power than h(), there is no horizontal asmptote. Let s investigate all three cases: a) f() = b) f() = + 3 c) f() = 4 d) 3+ e) = 3( ) 4 f) =

11 4.5B Rational Functions Part Eample Graph f() = +3 *To find the shape of the graph, it s helpful to choose values one less and one greater than vertical asmptote(s), and also values approaching both horizontal and vertical asmptotes. How would the graph be different if the function was g() = +3 4? Eample Graph =

12 Eample 3 Graph h() = + Eample 4 Graph f() = Eample 5 Graph = *A note about horizontal asmptotes:

4.6 Reciprocal Functions What is a reciprocal? 4 The following points are on the vertical number line: 4, 3,, Plot and label their reciprocals: 3 The reciprocal of is.

13 4.6 Reciprocal Functions What is a reciprocal? 4 The following points are on the vertical number line: 4, 3,, Plot and label their reciprocals: 3 The reciprocal of is. As the numbers increase towards infinit, their reciprocals 0 For negative numbers: The reciprocal of - is. As the numbers increase towards negative infinit, their reciprocals What is the reciprocal of 00? What is the reciprocal of ? As numbers decrease toward zero, how do their reciprocals behave? As negative numbers increase toward zero, how do their reciprocals behave?

14 Eample Graph = and its reciprocal on the same coordinate plane = = What is unique about the reciprocals of - and in this eample? In this eample, (, ) and (-, -) are called invariant points, as the are the same for the original and reciprocal. Invariant points are alwas where f() = ± Before plotting points on the graph, let s find an vertical and horizontal asmptotes for = using methods learned in section 4.5A. Vertical Asmptote: Horizontal Asmptote: Notice that the -intercept on the original function becomes the vertical asmptote on the reciprocal function. Remember, the -intercept is where, or f() is equal to zero, so when this becomes a reciprocal, f(), the reciprocal function is now undefined.

15 Eample Graph = 4 and then graph its reciprocal b first identifing an asmptotes, and then identifing invariant points. Eample 3 Draw the graph of the reciprocal function using the original function graph.

REVIEW, pages

REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in