Problem 112 Graph each function. $$D(t)=3 ... [FREE SOLUTION]

Chapter 5: Problem 112

Graph each function. $$D(t)=3 t$$

Short Answer

Expert verified

To graph the linear function $D(t) = 3t$, identify the slope (3) and y-intercept (0) from the equation. Plot the y-intercept point (0,0) and at least two more points utilizing the slope. For example, (1,3) and (2,6). Finally, draw a straight line through these points to represent the function $D(t) = 3t$.

Step by step solution

Identify the type of function

In this exercise, we are given the function D(t) = 3t. This is a linear function, as the variable t is raised to the power of 1. A linear function represents a straight line on the graph when plotted.

Find the slope and y-intercept of the function

The given function is already in the slope-intercept form, which is: $$y = mx + b$$ Where m is the slope and b is the y-intercept. In our case: $$D(t) = 3t$$ Comparing the given function to the slope-intercept form, we find that the slope (m) is 3 and the y-intercept (b) is 0.

Plot the y-intercept

The y-intercept occurs when the input (t) is 0. In this case, since b is 0, the y-intercept is at the point (0,0). Plot this point on the graph.

Plot two more points using the slope

Since the slope is 3, this means that for every increase of 1 unit in t (horizontal direction), the output D(t) will increase by 3 units (vertical direction). From the y-intercept (0,0), we can find more points by applying the slope: 1. Move 1 unit to the right along the t-axis (increase t by 1), and then move 3 units upward along the D(t)-axis (increase D(t) by 3). This gives us the point (1,3). 2. Move 1 more unit to the right along the t-axis (increase t by 1), and then move 3 units upward along the D(t)-axis (increase D(t) by 3). This gives us the point (2,6). Plot these points on the graph.

Draw the line

Now that we have plotted three points on the graph, we can draw a straight line through these points. The line represents the function D(t) = 3t.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form

Linear equations are often written in the slope-intercept form, which is an incredibly useful way to express a line mathematically. This form makes it easy to identify the key components of a line: its slope and its y-intercept. The general expression for a line in the slope-intercept form is $y = mx + b$, where:

$m$ represents the slope.
$b$ represents the y-intercept.

In the function $D(t) = 3t$, we can see it naturally fits the slope-intercept form even without a separate "b" term. Here, the slope $m$ is 3, and the y-intercept $b$ is 0. The function of this form is vital as it simplifies understanding and graphing linear equations, especially for quick calculation or estimation.

Graphing Linear Equations

Graphing a linear equation involves plotting points that lie on the line and then connecting them with a straight edge. It's a straightforward process if we follow these steps:

First, identify known points like the y-intercept and other easy-to-calculate points using the slope.
Start by marking the y-intercept on the graph, the point where the line crosses the y-axis. For $D(t) = 3t$, this point is (0,0).
Use the slope to identify more points on the line by moving along the axis. For a slope of 3, you rise 3 units vertically for each step of 1 unit horizontally.

For example, starting from the y-intercept (0,0), use the slope to locate points like (1,3) and (2,6). Once these points are plotted on the graph, simply draw a line through the points to visualize the function clearly. It鈥檚 essential to ensure the line is straight and extends well across the graph to capture the trend of the equation.

Slope and Y-Intercept

Understanding the slope and y-intercept is key to mastering linear equations.

The slope tells us how steep the line is and the direction it鈥檚 heading. A positive slope, like 3 in $D(t) = 3t$, means the line ascends from left to right.
The slope is calculated as the ratio of the rise (vertical change) to the run (horizontal change). For each unit increase in $t$, $D(t)$ increases by 3 units.
The y-intercept is the point at which the line crosses the y-axis. This happens when the value of $t$ is zero.

In our example, the y-intercept $b$ is 0, indicating the line passes through the origin. When graphing, the y-intercept is typically your starting point since it is easily identifiable. From there, using the slope, you can find additional points to draw the entire line of the function. The simplicity of plotting with the slope and y-intercept makes it easier to understand how the function behaves without complex calculations.

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91影视

Graph each function. $$D(t)=3 t$$

Short Answer

Step by step solution

Identify the type of function

Find the slope and y-intercept of the function

Plot the y-intercept

Plot two more points using the slope

Draw the line

Key Concepts

Slope-Intercept Form

Graphing Linear Equations

Slope and Y-Intercept

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