The maximum benefit for the heart from exercising occurs if...

The maximum benefit for the heart from exercising occurs if the heart rate is in the target heart rate zone. The lower limit of this target zone can be approximated by the linear equation $$ y=-0.65 x+143 $$ where $x$ represents age and $y$ represents heartbeats per minute. (Source: The Gazette.) (a) Complete the table of values for this linear equation. (b) Write the data from the table of values as ordered pairs. (c) Make a scatter diagram of the data. Do the points lie in an approximately linear pattern?

The maximum benefit for the heart from exercising occurs if the heart rate is in the target heart rate zone. The lower limit of this target zone can be approximated by the linear equation
$$
y=-0.65 x+143
$$
where $x$ represents age and $y$ represents heartbeats per minute. (Source: The Gazette.)
(a) Complete the table of values for this linear equation.
(b) Write the data from the table of values as ordered pairs.
(c) Make a scatter diagram of the data. Do the points lie in an approximately linear pattern?

Key Concepts

Scatter Diagrams

A scatter diagram is a graphical tool used to plot ordered pairs on a coordinate plane to visually assess the relationship between the variables. By looking at the pattern of the points, one can determine whether the relationship is linear or follows another type of pattern.

Linear Equations

A linear equation represents a straight-line relationship between two variables, commonly written in the form y = mx + b, where m is the slope and b is the y-intercept. This form allows one to predict or calculate one variable given the other and understand how changes in one variable affect the other.

Function Evaluation and Table of Values

Creating a table of values involves substituting different input values (often represented on the x-axis) into the given linear equation to calculate and record the corresponding output values (y). This process helps in understanding the behavior of the function over a range of inputs and is essential for plotting the function accurately.

Ordered Pairs

Ordered pairs are used to represent the relationship between two variables in a coordinate system, typically written as (x, y). Each pair corresponds to a specific value of the input variable and its resulting output value from the equation, forming the basis for graphing the function.

Linear Pattern

When the points on a scatter diagram lie approximately along a straight line, it suggests a linear relationship between the variables. This linearity means that the rate of change between the variables remains consistent, which is often indicative of a predictable and proportional relationship.

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