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Absolute Value And Step Functions Common Core Algebra 1 Homework Answers



Absolute Value And Step Functions Common Core Algebra 1 Homework Answers




Are you looking for absolute value and step functions common core algebra 1 homework answers? If so, you have come to the right place. In this article, we will explain what absolute value and step functions are, how to graph them, how to solve equations and inequalities involving them, and how to use them in word problems. We will also provide some examples and exercises for you to practice your skills and check your understanding.




Absolute Value And Step Functions Common Core Algebra 1 Homework Answers



What are Absolute Value and Step Functions?




Absolute value is a mathematical concept that measures the distance of a number from zero on the number line. For example, the absolute value of -5 is 5, because -5 is 5 units away from zero. The absolute value of 3 is also 3, because 3 is also 3 units away from zero. The absolute value of zero is zero, because zero is zero units away from zero. The symbol for absolute value is , so we write -5 = 5, 3 = 3, and 0 = 0.


A step function is a function that has a constant value for each interval of its domain. For example, the greatest integer function is a step function that rounds down any number to the nearest integer. The graph of this function looks like a staircase, with horizontal steps at each integer value. The notation for this function is [ x ], so we write [ 2.7 ] = 2, [ -1.4 ] = -2, and [ 0 ] = 0.


How to Graph Absolute Value and Step Functions?




To graph an absolute value function, we need to remember that the shape of the graph is a V or an inverted V, depending on whether the function is positive or negative. The vertex of the graph is the point where the function changes direction, and it corresponds to the value of x that makes the expression inside the absolute value equal to zero. The slope of the graph is determined by the coefficient of x inside the absolute value.


For example, consider the function f(x) = x - 2 + 1. To graph this function, we first find the vertex by setting x - 2 = 0 and solving for x. We get x = 2, so the vertex is (2, 1). Then we find the slope by looking at the coefficient of x inside the absolute value. In this case, it is 1, so the slope is 1 on both sides of the vertex. We can use these information to plot some points and draw the graph as shown below.


To graph a step function, we need to remember that the graph consists of horizontal line segments that jump from one value to another at each integer point. The value of the function at each integer point depends on whether we use a left-bracket [ or a right-bracket ] in the notation. A left-bracket means that we include the integer point in the interval, while a right-bracket means that we exclude it.


For example, consider the function g(x) = [ x ]. To graph this function, we first find the value of g(x) at each integer point by rounding down x to the nearest integer. For example, g(0) = [ 0 ] = 0, g(1) = [ 1 ] = 1, g(2) = [ 2 ] = 2, g(-1) = [ -1 ] = -1, g(-2) = [ -2 ] = -2, and so on. Then we draw horizontal line segments that connect these points and jump at each integer point as shown below.


How to Solve Equations and Inequalities Involving Absolute Value and Step Functions?




To solve equations involving absolute value, we need to remember that there are two possible cases: either the expression inside the absolute value is equal to


How to Use Absolute Value and Step Functions in Word Problems?




One of the most common applications of absolute value and step functions is in word problems that involve distance, speed, time, temperature, or other quantities that can be positive or negative. To solve these problems, we need to translate the words into mathematical expressions, equations, or inequalities, and then use the properties and methods of absolute value and step functions to find the solution.


For example, suppose we want to find the distance between two points on a number line, A and B. We can use the absolute value function to find the distance by subtracting the coordinates of A and B and taking the absolute value of the result. For instance, if A is at -3 and B is at 5, then the distance between them is -3 - 5 = -8 = 8 units.


Another example is to find the speed of a car that travels a certain distance in a certain time. We can use the step function to find the speed by dividing the distance by the time and rounding down to the nearest integer. For instance, if a car travels 120 miles in 2 hours, then its speed is [ 120 / 2 ] = [ 60 ] = 60 miles per hour.


Examples and Exercises




Let's look at some examples and exercises of absolute value and step functions common core algebra 1 homework answers.


Example 1




Graph the function h(x) = - x + 1 - 2.


Solution:


To graph this function, we first find the vertex by setting x + 1 = 0 and solving for x. We get x = -1, so the vertex is (-1, -2). Then we find the slope by looking at the coefficient of x inside the absolute value. In this case, it is -1, so the slope is -1 on both sides of the vertex. We can use these information to plot some points and draw the graph as shown below.


Example 2




Solve the equation 2x - 5 = 9.


Solution:


To solve this equation, we need to remember that there are two possible cases: either 2x - 5 = 9 or 2x - 5 = -9. We can solve for x in each case and check if the solution satisfies the original equation.


Case 1: 2x - 5 = 9


Add 5 to both sides: 2x = 14


Divide both sides by 2: x = 7


Check: 2(7) - 5 = 14 - 5 = 9 = 9 ️


Case 2: 2x - 5 = -9


Add 5 to both sides: 2x = -4


Divide both sides by 2: x = -2


Check: 2(-2) - 5 = -4 - 5 = -9 = 9 ️


Therefore, the solutions are x = 7 and x = -2.


Example 3




Solve the inequality [ x + 3 ]


Solution:


To solve this inequality, we need to remember that the step function [ x + 3 ] has a constant value for each interval of its domain. The intervals are determined by the integer points where x + 3 is equal to an integer. For example, when x + 3 = -4, x = -7; when x + 3 = -3, x = -6; when x + 3 = -2, x = -5; and so on. We can use these intervals to test the inequality and find the solution set.


Interval: (-, -7)


Pick a test point: x = -8


Plug in: [ (-8) + 3 ]


Simplify: [ -5 ]


Evaluate: False


Interval: [-7, -6)


Pick a test point: x = -6.5


Plug in: [ (-6.5) + 3 ]


Simplify: [ -3.5 ]


Evaluate: False


Interval: [-6, -5)


Pick a test point: x = -5.5


Plug in: [ (-5.5) + 3 ]


Simplify: [ -2.5 ]


Evaluate: True ️


Interval: [-5, )


Pick a test point: x = -4


Plug in: [ (-4) + 3 ]


Simplify: [ -1 ]


Evaluate: False


Therefore, the solution set is [-6, -5).


Example 4




Graph the function k(x) = [ x - 2 ] + 3.


Solution:


To graph this function, we first find the value of k(x) at each integer point by rounding down x - 2 to the nearest integer and adding 3. For example, k(0) = [ 0 - 2 ] + 3 = [ -2 ] + 3 = -2 + 3 = 1, k(1) = [ 1 - 2 ] + 3 = [ -1 ] + 3 = -1 + 3 = 2, k(2) = [ 2 - 2 ] + 3 = [ 0 ] + 3 = 0 + 3 = 3, k(3) = [ 3 - 2 ] + 3 = [ 1 ] + 3 = 1 + 3 = 4, and so on. Then we draw horizontal line segments that connect these points and jump at each integer point as shown below.


Exercise




Try to solve the following problems on your own. You can check your answers with the solutions provided at the end of this article.


  • Solve the equation x + 4 = 7.



  • Solve the inequality [ x - 5 ] > -2.



  • Graph the function f(x) = x + 3 - 4.



  • Graph the function g(x) = [ x / 2 ].



  • A car rental company charges $25 per day plus $0.15 per mile for renting a car. Write a step function that models the total cost of renting a car for a day as a function of the number of miles driven.



A thermometer measures the temperature in degrees Celsius. The absolute value function can be used to find the difference between two temperatures. For example, -10 - (-5) = -10 + 5 = -5 = 5, so the difference between -10C and -5C is


  • A thermometer measures the temperature in degrees Celsius. The absolute value function can be used to find the difference between two temperatures. For example, -10 - (-5) = -10 + 5 = -5 = 5, so the difference between -10C and -5C is 5C. Write an absolute value equation that can be used to find the difference between any two temperatures.



  • A parking garage charges $3 for the first hour and $2 for each additional hour or fraction thereof. Write a step function that models the total cost of parking as a function of the number of hours.



Solutions




  • Solve the equation x + 4 = 7.



  • There are two possible cases: either x + 4 = 7 or x + 4 = -7. We can solve for x in each case and check if the solution satisfies the original equation.



  • Case 1: x + 4 = 7



  • Subtract 4 from both sides: x = 3



  • Check: 3 + 4 = 7 = 7 ️



  • Case 2: x + 4 = -7



  • Subtract 4 from both sides: x = -11



  • Check: -11 + 4 = -7 = 7 ️



  • Therefore, the solutions are x = 3 and x = -11.



  • Solve the inequality [ x - 5 ] > -2.



  • We need to remember that the step function [ x - 5 ] has a constant value for each interval of its domain. The intervals are determined by the integer points where x - 5 is equal to an integer. For example, when x - 5 = -3, x = 2; when x - 5 = -2, x = 3; when x - 5 = -1, x = 4; and so on. We can use these intervals to test the inequality and find the solution set.



  • Interval: (-, 2)



  • Pick a test point: x = 1



  • Plug in: [ (1) - 5 ] > -2



  • Simplify: [ -4 ] > -2



  • Evaluate: False



  • Interval: [2, 3)



  • Pick a test point: x = 2.5



  • Plug in: [ (2.5) - 5 ] > -2



  • Simplify: [ -2.5 ] > -2



  • Evaluate: False



  • Interval: [3, )



  • Pick a test point: x = 4



  • Plug in: [ (4) - 5 ] > -2



  • Simplify: [ -1 ] > -2



  • Evaluate: True ️



  • Therefore, the solution set is [3, ).



Graph the function f(x) = x + 3 -


Conclusion




In this article, we have learned about absolute value and step functions common core algebra 1 homework answers. We have explained what absolute value and step functions are, how to graph them, how to solve equations and inequalities involving them, and how to use them in word problems. We have also provided some examples and exercises for you to practice your skills and check your understanding.


Absolute value and step functions are important concepts in algebra that can help us model and solve real-world problems involving distance, speed, time, temperature, or other quantities that can be positive or negative. By mastering these topics, you will be able to apply them to various situations and scenarios in mathematics and beyond.


We hope you have enjoyed this article and found it helpful. If you have any questions or feedback, please feel free to leave a comment below. Thank you for reading and happy learning! 6c859133af


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