MTH120 College Algebra Chapter 3.6

 3.6 Absolute Value Functions:

An absolute value function is a mathematical function that calculates the absolute value of its input, denoted as |x|, which is defined as follows:

For any real number x:

• If x is positive or zero, |x| = x.
• If x is negative, |x| = -x.

In other words, the absolute value of a number is its distance from zero on the number line, and it is always a non-negative (positive or zero) value.

The graph of an absolute value function typically has a "V" shape. Here is a general form for an absolute value function:

f(x) = |x - h| + k

In this equation:

• f(x) represents the output (or the function value) of the absolute value function for a given input x.
• x is the input (the variable you want to find the absolute value of).
• h and k are constants that shift the graph horizontally and vertically, respectively.

The graph of f(x) = |x - h| + k is a translation of the basic absolute value function |x|, with the graph shifted horizontally h units to the right (if h is positive) or to the left (if h is negative) and vertically k units up (if k is positive) or down (if k is negative).

For example:

• The function f(x) = |x| is the basic absolute value function with no horizontal or vertical shifts, so its graph is a V-shape centered at the origin.
• The function f(x) = |x - 2| shifts the graph of |x| two units to the right.
• The function f(x) = |x + 3| + 1 shifts the graph of |x| three units to the left and one unit up.

Absolute value functions are used in various mathematical and real-world applications, including solving equations involving absolute values, modeling situations with distance or magnitude, and optimizing problems where you want to minimize or maximize a value subject to certain constraints.

Understanding absolute value is fundamental in mathematics, particularly in real numbers and algebra. Absolute value is a measure of the "distance" of a number from zero on the number line, and it always yields a non-negative (positive or zero) result. Here are the key concepts to help you understand absolute value:

1. Definition: The absolute value of a real number x, denoted as |x|, is defined as follows:

   • If x is greater than or equal to zero (x ≥ 0), then |x| = x.
   • If x is less than zero (x < 0), then |x| = -x.

   In other words, |x| "absolutely" ignores the sign of x and gives you the positive magnitude of x.

2. Visual Representation: On the number line, the absolute value of a number is the distance of that number from zero, regardless of whether it is to the left (negative direction) or to the right (positive direction) of zero. For example:

   • |3| = 3 because 3 units to the right of zero.
   • |-5| = 5 because 5 units to the left of zero.

   This geometric interpretation helps in understanding absolute value as a measure of distance.

3. Properties of Absolute Value:

   • |x| is always greater than or equal to zero: |x| ≥ 0.
   • |x| = 0 if and only if x = 0.
   • |x| = |-x|, meaning the absolute value is symmetric with respect to the origin.

4. Applications:

   • Solving Equations: Absolute value often appears in equations, and solving absolute value equations involves considering both the positive and negative possibilities. For example, solving |x - 2| = 3 yields two solutions: x = 5 and x = -1.

   • Distance: Absolute value is used to calculate distances between points in various contexts, such as geometry or physics. The distance between two numbers a and b on the number line is given by |a - b|.

   • Inequalities: Absolute value is useful in solving inequalities involving expressions with absolute values. These inequalities often have two cases to consider based on the sign of the expression inside the absolute value.

5. Notation: In mathematical notation, the absolute value of a number is represented using vertical bars, such as |x|. Some calculators and programming languages use functions like "abs(x)" to compute the absolute value.

In summary, absolute value is a mathematical concept that measures the distance of a number from zero, regardless of its sign. It has practical applications in various mathematical and real-world scenarios, including equations, distances, and inequalities. Understanding absolute value is crucial for a solid foundation in mathematics.

An absolute value function is a mathematical function that calculates the absolute value of its input. The absolute value of a real number x, denoted as |x|, is defined as follows:

• If x is greater than or equal to zero (x ≥ 0), then |x| = x.
• If x is less than zero (x < 0), then |x| = -x.

In essence, the absolute value function takes a real number as input and returns its positive value (if it's non-negative already) or negates it (if it's negative).

The general form of an absolute value function is:

f(x) = |x|

This is the simplest absolute value function, which graphs as a V-shaped curve centered at the origin (0,0) on the Cartesian plane. The graph is a straight line with a slope of 1 on the right side of the origin and a slope of -1 on the left side.

Here are some properties and characteristics of absolute value functions:

1. Symmetry: The graph of an absolute value function is symmetric with respect to the y-axis. This means that if you replace x with -x in the function, the function value remains the same: |x| = |-x|.

2. Range: The range of an absolute value function is all non-negative real numbers, which means that the output values (f(x)) are always greater than or equal to zero.

3. Minimum Value: The absolute value function has a minimum value of 0 at the vertex (0, 0). This is the point where the graph changes direction from decreasing to increasing.

4. Transformation: You can transform the basic absolute value function, |x|, by adding constants to it or multiplying it by constants. For example, f(x) = 2|x - 3| + 1 represents a transformation of |x|, shifting it three units to the right and one unit up.

5. Solving Equations: Absolute value functions are often used to solve equations involving absolute values. These equations typically have two cases to consider, one for the positive value inside the absolute value bars and another for the negative value.

6. Applications: Absolute value functions have practical applications in various fields, including physics, engineering, economics, and computer science. They are used to model situations where the magnitude of a quantity matters more than its sign.

Overall, absolute value functions are fundamental in mathematics and find widespread use in modeling real-world scenarios and solving equations involving absolute values.

Graphing an absolute value function involves plotting points on the Cartesian plane to visualize how the function behaves. The general form of an absolute value function is:

f(x) = |x|

This is the simplest absolute value function, and its graph is a V-shaped curve centered at the origin (0,0). Here's how you can graph this basic absolute value function:

1. Choose some x-values: Decide on a range of x-values you want to plot on the graph. You can start with a few values around zero, both positive and negative.

2. Calculate the corresponding y-values: Use the absolute value function f(x) = |x| to find the corresponding y-values (f(x)) for each x-value you chose. Since |x| simply gives the positive value of x, this is straightforward. For example:

   • If x = -2, then f(-2) = |-2| = 2.
   • If x = 0, then f(0) = |0| = 0.
   • If x = 2, then f(2) = |2| = 2.

3. Plot the points: Plot the points (x, f(x)) on the Cartesian plane. For the examples above, you would plot (-2, 2), (0, 0), and (2, 2).

4. Connect the points: Draw a smooth curve that passes through the plotted points. Since the absolute value function is symmetric, your graph should look like a V-shape centered at the origin.

5. Label the axes: Make sure to label the x and y-axes and add a title to your graph.

6. Extend the graph: You can extend the graph in both the positive and negative directions to show that the function continues infinitely in both directions.In this graph, the V-shaped curve is centered at the origin, and it extends infinitely in both the positive and negative directions along the x-axis.

   You can also graph more complex absolute value functions by applying transformations like shifting the graph horizontally or vertically or stretching/compressing it, but the basic steps of choosing x-values, calculating y-values, plotting points, and connecting them remain the same.

7. If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

8. 
   

9. Yes, you can algebraically determine the stretch (or compression) of an absolute value function by examining its equation. The stretch or compression of an absolute value function is related to a coefficient applied to the absolute value expression, and you can identify it by analyzing the coefficients in the equation. Here's how:

   The general form of an absolute value function is:

   f(x) = a |x - h| + k

   In this equation:

   • "a" represents the coefficient that determines the stretch or compression. If |a| > 1, it indicates a vertical stretch, while if 0 < |a| < 1, it indicates a vertical compression.

   • "h" represents a horizontal shift (a shift left or right) of the graph.

   • "k" represents a vertical shift (a shift up or down) of the graph.

   To determine the stretch or compression:

   1. If |a| > 1: This indicates a vertical stretch. The larger the absolute value of "a," the greater the stretch. For example, if a = 2, the graph is stretched vertically by a factor of 2.

   2. If 0 < |a| < 1: This indicates a vertical compression. The smaller the absolute value of "a," the greater the compression. For example, if a = 0.5, the graph is compressed vertically by a factor of 0.5.

   3. If |a| = 1: This means there is no vertical stretch or compression. The graph remains the same as the basic absolute value function |x|.

   So, by examining the value of "a" in the equation, you can determine whether the function is stretched, compressed, or not stretched at all vertically.

10. Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

11. The graphs of absolute value functions do not always intersect the vertical axis, but they always intersect the horizontal axis. Let's discuss each of these cases:

    1. Intersection with the Vertical Axis:

       • The vertical axis corresponds to the line x = 0 on the Cartesian plane. For absolute value functions in the form of f(x) = |x - h| + k, where "h" and "k" are constants, the graph intersects the vertical axis (x = 0) if and only if h = 0.
       • In simpler terms, if the absolute value function is of the form f(x) = |x| or f(x) = |x + k|, where k is a constant, then the graph intersects the vertical axis at x = 0. This is because the absolute value of x is always defined at x = 0.
       • However, if you have a horizontal shift in the function, such as f(x) = |x - 3|, the graph will intersect the vertical axis at x = 3.

    2. Intersection with the Horizontal Axis:

       • The horizontal axis corresponds to the line y = 0 on the Cartesian plane. For all absolute value functions, the graph always intersects the horizontal axis (y = 0) at the origin, which is the point (0, 0).
       • This is because the absolute value function f(x) = |x| is defined as |x| = 0 when x = 0, which means that the graph passes through the origin.

    In summary, the graphs of absolute value functions always intersect the horizontal axis at the origin (0, 0) because the absolute value of 0 is 0. However, whether they intersect the vertical axis (x = 0) depends on whether there is a horizontal shift (h) in the function. If there is no horizontal shift (h = 0), the graph intersects the vertical axis at x = 0. If there is a horizontal shift (h ≠ 0), the intersection point will be at x = h.

Solving an absolute value equation involves finding the values of the variable that satisfy the equation. Absolute value equations often have two solutions: one for the positive case and one for the negative case. Here's a general approach to solving an absolute value equation:

1. Isolate the Absolute Value Expression: Start by isolating the absolute value expression on one side of the equation. For example, if you have the equation:

   |2x - 1| = 5

   You would write it as:

   2x - 1 = 5 (for the positive case) and 2x - 1 = -5 (for the negative case)

2. Solve for the Variable in Each Case:

   • For the positive case: Solve the equation as if there were no absolute value symbols. In this example, you would add 1 to both sides and then divide by 2:

     2x - 1 = 5 2x = 6 x = 3

   • For the negative case: Again, solve the equation as if there were no absolute value symbols, but remember to negate the right-hand side of the equation:

     2x - 1 = -5 2x = -4 x = -2

3. Write Down Both Solutions: Since absolute value equations typically have two solutions, you should write down both solutions:

   For the positive case: x = 3 For the negative case: x = -2

4. Check Solutions: It's essential to check your solutions by substituting them back into the original equation to ensure they satisfy it. In this case:

   • For x = 3: |2(3) - 1| = 5 |6 - 1| = 5 |5| = 5 (True)

   • For x = -2: |2(-2) - 1| = 5 |-4 - 1| = 5 |-5| = 5 (True)

Both solutions x = 3 and x = -2 satisfy the original equation, so they are both valid solutions.

Remember that not all absolute value equations have two solutions. Some equations may have one solution or no real solutions, depending on the specific equation and the values involved. Always check your solutions to ensure they are valid for the original equation.

Let's go through some examples of solving absolute value equations:

Example 1: Solve the equation |3x + 2| = 7.

1. Isolate the absolute value expression: 3x + 2 = 7 (for the positive case) and 3x + 2 = -7 (for the negative case)

2. Solve for the variable in each case:

   • For the positive case: 3x + 2 = 7 3x = 5 x = 5/3

   • For the negative case: 3x + 2 = -7 3x = -9 x = -3

3. Write down both solutions: For the positive case: x = 5/3 For the negative case: x = -3

4. Check solutions:

   • For x = 5/3: |3(5/3) + 2| = 7 |5 + 2| = 7 |7| = 7 (True)

   • For x = -3: |3(-3) + 2| = 7 |-9 + 2| = 7 |-7| = 7 (True)

Both solutions x = 5/3 and x = -3 are valid solutions.

Example 2: Solve the equation |2x - 4| = 3.

1. Isolate the absolute value expression: 2x - 4 = 3 (for the positive case) and 2x - 4 = -3 (for the negative case)

2. Solve for the variable in each case:

   • For the positive case: 2x - 4 = 3 2x = 7 x = 7/2

   • For the negative case: 2x - 4 = -3 2x = 1 x = 1/2

3. Write down both solutions: For the positive case: x = 7/2 For the negative case: x = 1/2

4. Check solutions:

   • For x = 7/2: |2(7/2) - 4| = 3 |7 - 4| = 3 |3| = 3 (True)

   • For x = 1/2: |2(1/2) - 4| = 3 |1 - 4| = 3 |-3| = 3 (True)

Both solutions x = 7/2 and x = 1/2 are valid solutions.

These examples illustrate how to solve absolute value equations by considering both the positive and negative cases and checking the solutions in the original equations to confirm their validity.

Absolute value functions and equations have numerous real-world applications in various fields, including physics, engineering, economics, and computer science. Here are some examples of real-world applications along with specific scenarios:

1. Distance and Speed:

   • Example: A car is traveling at a constant speed of 60 miles per hour. The distance, d, the car travels in t hours can be represented by the equation d(t) = 60t. If you want to know how far the car has traveled after 2 hours, you would calculate d(2) = 60 * 2 = 120 miles. In this case, the absolute value function isn't directly used, but the concept of distance and speed is fundamental in applications involving absolute value.

2. Temperature Control:

   • Example: A thermostat in a room is set to maintain a temperature of 72°F. If the actual room temperature drops below 72°F, the thermostat will activate the heater until the temperature reaches 72°F again. The difference between the set temperature and the actual temperature is represented by an absolute value equation: |Actual Temperature - 72| = Desired Temperature Range. The thermostat uses this equation to determine when to turn the heater on and off.

3. Stock Market:

   • Example: In finance, the absolute value function can be used to calculate the difference between the current stock price and a target price. For instance, if you own a stock currently trading at $50 per share, and you have a target selling price of $60 per share, you might track the absolute value of the difference |$50 - $60| = $10 to determine when to sell the stock.

4. Electrical Circuits:

   • Example: In electrical engineering, absolute value functions can represent components like diodes and certain types of sensors. Diodes allow electrical current to flow in one direction, essentially "rectifying" it. The voltage across a diode is an absolute value-like function as it ensures current flows in one direction only.

5. Signal Processing:

   • Example: In signal processing, absolute value is used to calculate the magnitude of signals, such as in audio processing to determine the loudness of a sound. The absolute value of a signal is often used to rectify the signal, making it always positive and suitable for further analysis.

6. Optimization Problems:

   • Example: In mathematical optimization problems, absolute value constraints can be used. For instance, in linear programming, absolute value constraints might appear in scenarios like optimizing transportation routes, where constraints ensure that the total distance traveled is within a certain range.

7. Robotics:

   • Example: In robotics and motion planning, absolute value functions can be used to calculate distances between objects in the environment. This information is crucial for collision detection and path planning to ensure that robots move safely.

8. Game Development:

   • Example: In video game programming, absolute value functions can be used to calculate distances between game objects or characters, determining their interactions or collisions.

These examples illustrate how absolute value functions and equations play a role in modeling, analyzing, and solving real-world problems across various domains. They are essential tools for understanding and solving problems involving magnitude, distance, difference, and constraints.