Problem 22 - Beginning and Intermediate Algebra

Question

Graph each function. $$r(t)=3 t-2$$

Step-by-Step Solution

Answer

To graph the function \(r(t) = 3t - 2\), we first choose at least two distinct values for \(t\), such as \(t=0\), \(t=1\), and \(t=2\). Then, we calculate the corresponding values of \(r(t)\) for each chosen value of \(t\). In this case, the points are \((0, -2)\), \((1, 1)\), and \((2, 4)\). Next, we plot these points on the Cartesian coordinate plane and connect them with a straight line. The graph of \(r(t) = 3t - 2\) is a straight line passing through the points \((0, -2)\), \((1, 1)\), and \((2, 4)\).

Step 1: Choose values for t

Choose at least two distinct values for \(t\). In this case, we will choose \(t=0\), \(t=1\), and \(t=2\).

Step 2: Calculate corresponding r(t) values

For each chosen value of \(t\), calculate the corresponding value of \(r(t)\) using the given formula \(r(t) = 3t - 2\). For \(t=0\): \(r(0) = 3(0) - 2 = -2\) For \(t=1\): \(r(1) = 3(1) - 2 = 1\) For \(t=2\): \(r(2) = 3(2) - 2 = 4\)

Step 3: Plot the points

Plot the points \((0, -2)\), \((1, 1)\), and \((2, 4)\) on the Cartesian coordinate plane.

Step 4: Connect the points

Draw a straight line that passes through all three points. The graph of the function \(r(t) = 3t - 2\) is a straight line, with the points \((0, -2)\), \((1, 1)\), and \((2, 4)\) lying on the line.

Key Concepts

Cartesian Coordinate Plane

The Cartesian coordinate plane is a fundamental concept in graphing. It's a flat surface defined by two axes, the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin. The origin has coordinates

• (0, 0)

Understanding this plane is essential, as it allows us to plot points and lines by locating their exact positions based on the x and y coordinates. Each point on the plane is represented by an ordered pair

• e.g.,
  (2, 3)

where the first number is the x-coordinate, indicating the horizontal position, and the second is the y-coordinate, indicating the vertical position.
When graphing functions, like in our exercise, we use this coordinate system to visually understand relationships.
In the given problem, points like (0, -2) or (2, 4) are plotted based on their respective t values, showing how the function behaves across this plane.

Linear Equations

Linear equations represent straight lines on the Cartesian coordinate plane. They take the general form:

• \( y = mx + b \)

where \(m\) is the slope and \(b\) is the y-intercept. In our exercise, though we use \(r(t)\) instead of \(y\), the equation \(r(t) = 3t - 2\) is a linear equation.
The number \(3\) is the slope, indicating the steepness and direction of the line.
This means that for every one unit increase in \(t\), \(r(t)\) increases by 3 units.
The \(-2\) is the y-intercept, which is the point where the line crosses the y-axis.
Understanding these components helps in quickly graphing the equation and predicting how changes in values affect the graph.

Graphing Points

Graphing points is a straightforward yet crucial step in visualizing equations. To graph a point in our example, you identify the coordinates derived from substituting values into the linear equation. For instance, when

• t=0

the calculated result for \(r(t) = -2\) gives the point (0, -2).

• For t=1

\(r(t) = 1\), resulting in (1, 1), and similarly,

• t=2

\(r(t) = 4\) results in (2, 4).
You locate these points on the Cartesian plane based on their x and y values.
Once all points are plotted, a straight line can be drawn through them.
This line is the graphical representation of the function, emphasizing the linearity and the relationship between the variables. Graphing such points accurately is key to creating an accurate visualization of any function.