Problem 53 - Algebra 1: Concepts and Skills

Question

Make an input-output table for the function \(A=8+5 t .\) Use \(2,3,4,5,\) and 6 as values for \(t\)

Step-by-Step Solution

Answer

The input-output table is: \n\n | t | A | \n |---|---| \n | 2 | 18 | \n | 3 | 23 | \n | 4 | 28 | \n | 5 | 33 | \n | 6 | 38 |

Step 1 Understand the Given Function

The function given is \(A=8+5t\), where 'A' is the result of the function for any value 't'. 't' is the independent variable and can be any real number. 'A' depends on 't', and changes as 't' changes. The constant '8' is where the line crosses the y-axis on a graph, and '5' is the slope of the line, so for each increase of 1 in 't', 'A' increases by 5.

Step 2 Create an Input-Output Table

A table with two columns can represent the function. The first column can have values of 't', and the corresponding values of 'A' will occupy the second column.

Step 3 Substitute Values

Now, use the provided values for 't' which are \(2,3,4,5,\) and \(6\) and substitute each into the function to compute 'A'. \n\n For \(t=2\), \(A=8+5*2=18\). \n For \(t=3\), \(A=8+5*3=23\). \n For \(t=4\), \(A=8+5*4=28\). \n For \(t=5\), \(A=8+5*5=33\). \n For \(t=6\), \(A=8+5*6=38\).

Step 4 Completing the Table

The completed input-output table is thus: \n\n | t | A | \n |---|---| \n | 2 | 18 | \n | 3 | 23 | \n | 4 | 28 | \n | 5 | 33 | \n | 6 | 38 |

Key Concepts

Linear Functions

Linear functions are a type of function that create straight lines when graphed. This is because they have a constant rate of change. The basic form of a linear function is \( y = mx + b \), where:

• \( m \) is the slope of the line, representing how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

For example, in the function \( A = 8 + 5t \), it is the same as \( y = 5t + 8 \). Here, \( 5 \) is the slope, meaning for each unit increase in \( t \), \( A \) increases by 5. The \( 8 \) is the y-intercept, where the line meets the y-axis when \( t \) is 0.
Linear functions are straightforward and predictable, making them easy to work with when creating tables or graphs.

Function Notation

Function notation is a way to express relationships between variables in a concise manner. It is commonly used in mathematics to show how one variable depends on another. It is often written as \( f(x) \), where:

• \( f \) represents the function name.
• \( x \) is the input or the independent variable.

For the given function \( A = 8 + 5t \), if we think of it in terms of function notation, it might look like \( f(t) = 8 + 5t \). This helps emphasize that 'A' is the value you get after applying the function rule to 't'.
This notation provides clarity especially when dealing with complex equations or when performing various operations like addition, subtraction, or composition of functions.

Independent and Dependent Variables

Understanding independent and dependent variables is crucial when studying functions.

• The independent variable, often represented by \( x \) or \( t \), is the variable that you change or control.
• The dependent variable, represented by \( y \) or \( f(x) \), is the result or outcome of the function based on the independent variable.

In our function \( A = 8 + 5t \), \( t \) is the independent variable. You choose different values for \( t \) like 2, 3, 4, 5, or 6. Based on these choices, \( A \) (the dependent variable) changes accordingly.
Understanding these roles helps you know what parts of the equation you can manipulate and what parts are outcomes of those manipulations.