If Mr. Garcia Works 20 Hours, He Will Receive $870. If He Works 40 Hours, He Will Receive $1,290. Write A Linear Function To Represent The Pay $p$, Based On Mr. Garcia Working $h$ Hours.

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Linear Function Representation of Mr. Garcia's Pay

In this article, we will explore the concept of linear functions and how they can be used to represent real-world scenarios. We will use the example of Mr. Garcia's pay to create a linear function that represents his earnings based on the number of hours he works.

Let's start by analyzing the given information:

  • If Mr. Garcia works 20 hours, he will receive $870.
  • If he works 40 hours, he will receive $1,290.

We can see that there is a direct relationship between the number of hours worked and the amount of pay received. This is a classic example of a linear relationship, where the output (pay) is directly proportional to the input (hours worked).

A linear function is a mathematical function that can be represented in the form:

p(h) = mh + b

where:

  • p(h) is the output (pay) for a given input (hours worked)
  • h is the input (hours worked)
  • m is the slope of the line (representing the rate of change)
  • b is the y-intercept (representing the starting point)

Our goal is to find the values of m and b that satisfy the given conditions.

To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

where:

  • y1 and y2 are the output values (pay) for the given input values (hours worked)
  • x1 and x2 are the input values (hours worked)

Plugging in the values, we get:

m = ($1,290 - $870) / (40 - 20) m = $420 / 20 m = $21

So, the slope (m) is $21.

Now that we have the slope, we can use one of the given conditions to find the y-intercept (b). Let's use the first condition:

p(20) = $870

Substituting the values into the linear function, we get:

$870 = 21(20) + b $870 = $420 + b b = $450

So, the y-intercept (b) is $450.

Now that we have the values of m and b, we can write the linear function:

p(h) = 21h + 450

This function represents Mr. Garcia's pay (p) based on the number of hours he works (h).

The linear function p(h) = 21h + 450 tells us that for every hour Mr. Garcia works, his pay increases by $21. The y-intercept of $450 represents the starting point, which is the amount he would receive if he worked 0 hours.

To visualize the linear function, we can graph it on a coordinate plane. The graph will be a straight line with a slope of $21 and a y-intercept of $450.

In this article, we used the example of Mr. Garcia's pay to create a linear function that represents his earnings based on the number of hours he works. We found the slope (m) and y-intercept (b) using the given conditions and wrote the linear function p(h) = 21h + 450. This function can be used to calculate Mr. Garcia's pay for any number of hours worked.

Linear functions have many real-world applications, including:

  • Modeling population growth
  • Representing the cost of goods and services
  • Calculating the area and perimeter of shapes
  • Determining the trajectory of objects in motion

By understanding and applying linear functions, we can make informed decisions and solve real-world problems.

In our previous article, we explored the concept of linear functions and how they can be used to represent real-world scenarios. We used the example of Mr. Garcia's pay to create a linear function that represents his earnings based on the number of hours he works. In this article, we will answer some frequently asked questions about linear functions.

A: A linear function is a mathematical function that can be represented in the form:

p(h) = mh + b

where:

  • p(h) is the output (pay) for a given input (hours worked)
  • h is the input (hours worked)
  • m is the slope of the line (representing the rate of change)
  • b is the y-intercept (representing the starting point)

A: The slope (m) of a linear function represents the rate of change of the output (pay) with respect to the input (hours worked). It is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where:

  • y1 and y2 are the output values (pay) for the given input values (hours worked)
  • x1 and x2 are the input values (hours worked)

A: The y-intercept (b) of a linear function represents the starting point of the line. It is the value of the output (pay) when the input (hours worked) is 0.

A: To find the slope (m) and y-intercept (b) of a linear function, you can use the following steps:

  1. Identify the input (hours worked) and output (pay) values.
  2. Use the formula m = (y2 - y1) / (x2 - x1) to calculate the slope (m).
  3. Use one of the given conditions to find the y-intercept (b).

A: Linear functions have many real-world applications, including:

  • Modeling population growth
  • Representing the cost of goods and services
  • Calculating the area and perimeter of shapes
  • Determining the trajectory of objects in motion

A: To graph a linear function, you can use the following steps:

  1. Identify the slope (m) and y-intercept (b) of the function.
  2. Plot the y-intercept (b) on the coordinate plane.
  3. Use the slope (m) to determine the direction and steepness of the line.
  4. Plot additional points on the line to create a graph.

A: Some common mistakes to avoid when working with linear functions include:

  • Confusing the slope (m) and y-intercept (b)
  • Failing to identify the input (hours worked) and output (pay) values
  • Using the wrong formula to calculate the slope (m)
  • Failing to check the units of the input and output values

In this article, we answered some frequently asked questions about linear functions. We covered topics such as the definition of a linear function, the slope (m) and y-intercept (b), and real-world applications. We also provided tips and common mistakes to avoid when working with linear functions. By understanding and applying linear functions, we can solve problems and make informed decisions in a variety of fields.

Linear functions are a powerful tool for representing real-world scenarios. By understanding the concept of linear functions and how to apply them, we can solve problems and make informed decisions. Whether it's modeling population growth or calculating the cost of goods and services, linear functions are an essential part of mathematics and have many real-world applications.