Problem 16 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=2 x\)

Step-by-Step Solution

Answer

Slope \( m = 2 \), \( y \)-intercept is (0, 0). Graph is a line through these points.

Step 1: Identifying the Equation Form

The given equation is in the form of a linear equation, specifically the slope-intercept form: \( y = mx + b \). In this form, \( m \) represents the slope and \( b \) represents the \( y \)-intercept.

Step 2: Finding the Slope \( m \)

In the equation \( y = 2x \), the coefficient of \( x \) is the slope \( m \). Thus, \( m = 2 \). The slope tells us that for every 1 unit increase in \( x \), \( y \) increases by 2 units.

Step 3: Finding the \( y \)-intercept \( b \)

In the equation \( y = 2x \), since there is no constant term added or subtracted, the \( y \)-intercept \( b \) is 0. This means the graph crosses the \( y \)-axis at the origin, (0, 0).

Step 4: Graphing the Equation

Start by plotting the \( y \)-intercept (0, 0) on the graph. Since the slope \( m = 2 \), which is \( \frac{2}{1} \), from the point (0, 0), move up 2 units and right 1 unit to locate another point on the graph (1, 2). Draw a straight line through these points to complete the graph.

Key Concepts

graphing linear equations

Graphing linear equations can seem complex at first, but it becomes straightforward once you understand the process. A linear equation describes a straight line on the coordinate plane. This is because the change between any two points on the line is uniform or constant. To graph a linear equation, we typically utilize information such as the slope and the y-intercept.
Simply follow these steps to graph a linear equation:

• Identify the slope and y-intercept from the equation. These will give you the starting point and direction in which to draw the line.
• Start by plotting the y-intercept on the graph. This is the point where the line crosses the y-axis.
• From the y-intercept, use the slope to find another point on the graph. The slope tells you how many units to move up or down for every unit you move left or right.
• Draw a line connecting these points. This line is the graphical representation of your equation.

By following these steps, graphing linear equations becomes a systematic and manageable task.

slope-intercept form

The slope-intercept form is one of the most common ways to represent linear equations. It is known for its simplicity and ease of use in graphing. The form is written as:

\( y = mx + b \)

Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.

• The **slope** (\( m \)) indicates how steep the line is. A larger absolute value of \( m \) means a steeper line. If \( m \) is positive, the line rises as you move from left to right; if negative, it falls.
• The **y-intercept** (\( b \)) is the point where the line crosses the y-axis. This is where \( x = 0 \).

The beauty of the slope-intercept form is that it directly provides both the slope and the y-intercept, making the task of graphing much simpler.

finding slope and y-intercept

Finding the slope and y-intercept from a linear equation is a crucial skill, especially when dealing with equations in the slope-intercept form.
Let's take the equation \( y = 2x \) as an example.

• **Finding the Slope**: In the equation \( y = mx + b \), the coefficient of \( x \) is the slope \( m \), which is 2 in this case. This means for every unit increase in \( x \), \( y \) will increase by 2 units.
• **Finding the Y-intercept**: The constant term in the equation represents the y-intercept \( b \). Since there is no constant term apart from \( 2x \), the y-intercept is \( b = 0 \). This indicates that the line will pass through the origin (0,0).

Recognizing these components allows you to quickly understand and graph the equation on a coordinate plane.