Problem 55 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of \(d=3 t+4\) using the horizontal axis for \(t\) values and the vertical axis for \(d\) values.

Step-by-Step Solution

Answer

Plot points (0, 4) and (1, 7), then draw a line through them.

Step 1: Identify the equation type

The given equation is a linear equation in the form of y = mx + b. Here, the equation is written as d = 3t + 4, where t is the independent variable and d is the dependent variable.

Step 2: Determine the slope and y-intercept

In the equation d = 3t + 4, the slope (m) is 3 and the y-intercept (b) is 4. This means that for every unit increase in t, d increases by 3 units, and the graph intersects the vertical axis at d = 4.

Step 3: Find two points on the line

To sketch the graph, find two points on the line. Start with t = 0: When t = 0, d = 3(0) + 4 = 4. So, one point is (0, 4). Now, choose another value for t, such as t = 1: When t = 1, d = 3(1) + 4 = 7. So, another point is (1, 7).

Step 4: Plot the points on the graph

On a graph with t-values on the horizontal axis and d-values on the vertical axis, plot the points (0, 4) and (1, 7).

Step 5: Draw the line

Draw a straight line through the points (0, 4) and (1, 7). This line represents the graph of the equation d = 3t + 4.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics. They describe relationships between two variables that create a straight line when graphed. The standard form of a linear equation is: \[y = mx + b\]. Here, 'y' is the dependent variable, 'x' is the independent variable, 'm' represents the slope, and 'b' stands for the y-intercept.
A linear equation means for every change in 'x', there is a consistent change in 'y'. Because of their simplicity and wide applicability, understanding linear equations is key for all math students.

Slope-Intercept Form

The slope-intercept form, \(y = mx + b\), is a common way to express linear equations. This form is incredibly useful for graphing because it tells you two crucial details immediately:

• Slope (m): This indicates the steepness of the line. For instance, in the equation \(d = 3t + 4\), the slope is 3, which means 'd' increases 3 units for every 1 unit increase in 't'.
• Y-intercept (b): This is the point where the line crosses the vertical axis. In our example, it's 4. So the line crosses the d-axis at (0,4).

The slope-intercept form helps you quickly understand how one variable affects another and makes plotting easier.

Plotting Points

Plotting points is the foundational step in graphing any equation. It begins with identifying key values for your variables. For the equation \(d=3t+4\), here’s how to do it:

• Choose Values: Select specific values for 't'. It's easiest to start with zero. For example, if \( t = 0 \), \( d = 3(0) + 4 = 4 \). This gives you the point (0, 4).
• Calculate Corresponding Values: Pick another value for 't', like 1. For \( t = 1 \), \( d = 3(1) + 4 = 7 \). This gives you another point (1, 7).
• Plot on Graph: On your graph paper or using diagram software, mark these points.
• Draw the Line: Connect these dots with a straight line. This line is the visual representation of your equation.

By plotting multiple points and drawing lines through them, you can effectively visualize how the variables interact.