Problem 80 - College Algebra

Question

For the following exercises, sketch the graph of each equation. $$ f(t)=3+2 t $$

Step-by-Step Solution

Answer

The graph of \(f(t) = 3 + 2t\) is a straight line with slope 2 and y-intercept 3.

Step 1: Identify the type of equation

Examine the given equation, \(f(t) = 3 + 2t\). This is a linear equation because it can be expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the slope and y-intercept

The slope \(m\) of the equation \(f(t) = 3 + 2t\) is 2, and the y-intercept \(b\) is 3. This means the line moves up 2 units vertically for each 1 unit it moves horizontally.

Step 3: Plot the y-intercept

On a graph, identify and mark the y-intercept. Since \(b = 3\), place a point at (0, 3) on the coordinate plane.

Step 4: Use the slope to find a second point

Starting from the y-intercept point (0, 3), use the slope to find another point. Since the slope is 2, move up 2 units and 1 unit to the right to reach the point (1, 5). Mark this point on the graph.

Step 5: Draw the line

With both points, (0, 3) and (1, 5), draw a straight line through these points. This line represents the graph of the equation \(f(t) = 3 + 2t\).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves creating a visual representation of an equation on a coordinate plane. It's like drawing a picture of the relationship between two variables. Consider the linear equation from the exercise, \(f(t) = 3 + 2t\). In a graph, this equation appears as a straight line because it is in the form \(y = mx + b\). Any equation in this form results in a straight line when plotted. Linear equations are foundational because they model relationships where two quantities maintain a constant rate of change. When graphing, you start by plotting known points and then connect them using a straight line. This shows all the possible solutions to the equation.

Slope and Y-Intercept

Understanding the slope and y-intercept makes graphing linear equations easier. The slope, denoted as \(m\) in \(y = mx + b\), indicates how steep the line is. In the equation \(f(t) = 3 + 2t\), the slope is 2. This means for every 1 unit of change in \(t\), \(f(t)\) changes by 2 units. It describes the rate of change and direction.

• If the slope is positive, the line rises from left to right.
• If it's negative, the line falls.

The y-intercept, represented by \(b\), is where the line crosses the y-axis. In our case, the y-intercept is 3, meaning the line crosses the y-axis at \((0, 3)\). This starting point helps you to begin plotting the graph of the line.

Coordinate Plane

The coordinate plane is a two-dimensional plane where you can plot points, lines, and curves to represent equations visually. It has two axes:

• The horizontal axis, or x-axis.
• The vertical axis, or y-axis.

These axes divide the plane into four quadrants. In the exercise, for the equation \(f(t) = 3 + 2t\), plotting starts by marking the y-intercept at \((0, 3)\) on the plane. Using the slope, you find another point like \((1, 5)\), and draw a line through these points. The coordinate plane lets you view how variable changes—like \(t\)—affect \(f(t)\), providing a clear picture of the linear relationship.