2 Unit Bridging Course Day 12
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1 1 / 38 2 Unit Bridging Course Day 12 Absolute values Clinton Boys
2 2 / 38 The number line The number line is a convenient way to represent all numbers: 0 We can put any number somewhere on this line (imagining it extends infinitely in both directions).
3 3 / 38 The number line The number line is a convenient way to represent all numbers: 0 We can put any number somewhere on this line (imagining it extends infinitely in both directions).
4 4 / 38 The number line The position of a number on the number line is completely specified by two pieces of information: (i) its magnitude or absolute value, that is, the absolute distance of the number from zero, and (ii) its sign, that is, whether it is positive (lies to the right of zero), or negative (lies to the left of zero).
5 5 / 38 The number line The position of a number on the number line is completely specified by two pieces of information: (i) its magnitude or absolute value, that is, the absolute distance of the number from zero, and (ii) its sign, that is, whether it is positive (lies to the right of zero), or negative (lies to the left of zero).
6 6 / 38 The number line The position of a number on the number line is completely specified by two pieces of information: (i) its magnitude or absolute value, that is, the absolute distance of the number from zero, and (ii) its sign, that is, whether it is positive (lies to the right of zero), or negative (lies to the left of zero).
7 7 / 38 Absolute value The absolute value of a number denotes how far away it is from zero, without regard to its sign. Thus the absolute value of a number x, which we write as x, is always positive (or zero if x = 0).
8 8 / 38 Absolute value The absolute value of a number denotes how far away it is from zero, without regard to its sign. Thus the absolute value of a number x, which we write as x, is always positive (or zero if x = 0).
9 9 / 38 Absolute value If x 0, the absolute value of x is just x itself since its sign is positive: x = x if x 0. However if x < 0, the absolute value of x is the positive number which sits on the other side of zero on the number line. This number is exactly the negative of the negative number x: x = x if x <
10 10 / 38 Absolute value If x 0, the absolute value of x is just x itself since its sign is positive: x = x if x 0. However if x < 0, the absolute value of x is the positive number which sits on the other side of zero on the number line. This number is exactly the negative of the negative number x: x = x if x <
11 11 / 38 Absolute value If x 0, the absolute value of x is just x itself since its sign is positive: x = x if x 0. However if x < 0, the absolute value of x is the positive number which sits on the other side of zero on the number line. This number is exactly the negative of the negative number x: x = x if x < = ( 2) = 2
12 12 / 38 Examples Example (i) 3 = 3 (ii) 4 = 4.
13 13 / 38 Equations involving absolute values Suppose we are given the equation x = 3 and asked to solve it for x. Since we now understand absolute values, we know there are exactly two possibilities: (1) x = 3 (2) x = 3.
14 14 / 38 Equations involving absolute values Suppose we are given the equation x = 3 and asked to solve it for x. Since we now understand absolute values, we know there are exactly two possibilities: (1) x = 3 (2) x = 3.
15 15 / 38 Equations involving absolute values Suppose we are given the equation x = 3 and asked to solve it for x. Since we now understand absolute values, we know there are exactly two possibilities: (1) x = 3 (2) x = 3.
16 16 / 38 Equations involving absolute values Suppose we are given the equation x = 3 and asked to solve it for x. Since we now understand absolute values, we know there are exactly two possibilities: (1) x = 3 (2) x = 3.
17 17 / 38 Equations involving absolute values We can use a similar strategy to solve general equations with absolute values: Example Solve the equation for x. x + 1 = 4 We know that if x + 1 = 4, then either x + 1 = 4, or x + 1 = 4.
18 18 / 38 Equations involving absolute values We can use a similar strategy to solve general equations with absolute values: Example Solve the equation for x. x + 1 = 4 We know that if x + 1 = 4, then either x + 1 = 4, or x + 1 = 4.
19 19 / 38 Equations involving absolute values So there are two possibilities for the solution: (1) x + 1 = 4, i.e. x = 3, or (2) x + 1 = 4, i.e. x = 5.
20 20 / 38 Equations involving absolute values So there are two possibilities for the solution: (1) x + 1 = 4, i.e. x = 3, or (2) x + 1 = 4, i.e. x = 5.
21 21 / 38 Equations involving absolute values So there are two possibilities for the solution: (1) x + 1 = 4, i.e. x = 3, or (2) x + 1 = 4, i.e. x = 5.
22 22 / 38 Equations involving absolute values Practice questions Find the values of x which satisfy the following equations: (i) x 2 = 4 (ii) 3x + 1 = 2 (iii) 2 + x = 1.
23 23 / 38 Equations involving absolute values Answers (i) x = 6 or x = 2 (ii) x = 1/3 or x = 1 (iii) x = 1 or x = 3.
24 24 / 38 Graphing absolute values Suppose we want to graph the function y = x. This function will look the same as the function y = x when x is positive, but when x is negative it will be the negative of y = x.
25 25 / 38 Graphing absolute values Suppose we want to graph the function y = x. This function will look the same as the function y = x when x is positive, but when x is negative it will be the negative of y = x.
26 26 / 38 Graphing absolute values Suppose we want to graph the function y = x. This function will look the same as the function y = x when x is positive, but when x is negative it will be the negative of y = x. y = x
27 27 / 38 Graphing absolute values Suppose we want to graph the function y = x. This function will look the same as the function y = x when x is positive, but when x is negative it will be the negative of y = x. y = x
28 28 / 38 Graphing absolute values This just corresponds to reflecting the negative part of the graph in the x-axis. Example Here are the graphs of y = 2x 1 and y = 2x 1.
29 29 / 38 Graphing absolute values This just corresponds to reflecting the negative part of the graph in the x-axis. Example Here are the graphs of y = 2x 1 and y = 2x 1.
30 30 / 38 Graphing absolute values This just corresponds to reflecting the negative part of the graph in the x-axis. Example Here are the graphs of y = 2x 1 and y = 2x 1. y = 2x 1
31 31 / 38 Graphing absolute values This just corresponds to reflecting the negative part of the graph in the x-axis. Example Here are the graphs of y = 2x 1 and y = 2x 1. y = 2x 1
32 32 / 38 Graphing absolute values Practice questions Sketch the following curves: (i) y = 3x 4 (ii) y = x 2 1 (iii) y = sin x
33 33 / 38 Answers (i) y = 3x 4 Graphing absolute values 4/3
34 34 / 38 Answers (i) y = 3x 4 Graphing absolute values 4/3
35 35 / 38 Graphing absolute values Answers (ii) y = x 2 1
36 36 / 38 Graphing absolute values Answers (ii) y = x 2 1
37 37 / 38 Graphing absolute values Answers (iii) y = sin x
38 38 / 38 Graphing absolute values Answers (iii) y = sin x
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