Problem 81 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Sketch a straight line with a slope of 3, intercepting the y-axis at -2.

Step 1: Identify the Equation Type

The given equation, \( p(t) = -2 + 3t \), is in the form of \( y = mx + b \). This means it is a linear equation which will graph as a straight line.

Step 2: Identify the Slope and Y-Intercept

In the equation \( p(t) = -2 + 3t \), the slope \( m \) is 3 and the y-intercept \( b \) is -2. This means the line will rise 3 units for each unit it moves to the right, and it crosses the y-axis at -2.

Step 3: Plot the Y-Intercept

To begin sketching the graph, plot the y-intercept. This is the point \( (0, -2) \) on the graph, meaning when \( t = 0 \), \( p(t) = -2 \).

Step 4: Use the Slope to Plot a Second Point

From the y-intercept, use the slope to find a second point. Since the slope is 3, move up 3 units and to the right 1 unit from \( (0, -2) \). This will take you to the point \( (1, 1) \). Plot this point.

Step 5: Draw the Line

Connect the two plotted points with a straight line. Extend the line in both directions, and make sure it's straight, indicating the linear nature of the equation.

Key Concepts

Graphing

Graphing is a visual representation of equations and their solutions on a coordinate plane. When we talk about graphing a linear equation, it's all about sketching a straight line. This line represents all solutions to the equation, visually showing how the variables relate to one another.
To graph a linear equation like our example, you'll follow a few essential steps:

• Identify crucial elements: This includes the slope and y-intercept, which guide all plotting actions.
• Plot the y-intercept: Begin your graph with the y-intercept point on the y-axis, as this sets the starting point of your line.
• Use slope to find another point: From the y-intercept, apply the slope to locate a second point, ensuring the line is accurately directioned.
• Connect the points: Draw a line through these points, extending it across the plane.

Graphing helps visualize abstract math problems, making them easier to understand and solve. It's key in determining relationships between variables and predicting outcomes based on data represented by the line.

Slope

The slope of a line quantifies the steepness and direction of the line on the graph. Represented by the letter \( m \) in the linear equation form \( y = mx + b \), the slope tells you how much the line rises or falls as you move from left to right.
For the equation \( p(t) = -2 + 3t \), the slope is 3, which means:

• The line rises 3 units vertically for every 1 unit it moves horizontally.
• This positive value indicates an upward tilt as you read from left to right.

Slopes can be positive, negative, zero, or undefined:

• Positive slope: Line ascends as we move right.
• Negative slope: Line descends as we move right.
• Zero slope: Line is flat, indicating no rise. Example equation \( y = b \).
• Undefined slope: Vertical line without horizontal movement. Example equation \( x = a \).

The slope is essential in predicting the change and directionality of a function, making it an important tool for any mathematical analysis.

Y-Intercept

The y-intercept is where the line crosses the y-axis, providing the starting point on the graph for plotting the equation. In the equation \( p(t) = -2 + 3t \), the y-intercept \( b \) is -2. This means that when \( t = 0 \), \( p(t) \) is -2.
Identifying the y-intercept is straightforward:

• It represents the value of the function when the independent variable is zero.
• This point on the graph is where you'd begin plotting the line.

The y-intercept offers a visual cue for beginning the graphing process, allowing you to determine another point using the slope. Essentially, it's the anchor or fixed point in the graphing process. Understanding the y-intercept allows for more accurate graphing and comprehension of the function's behavior in a real-life context.