Problem 45 - Calculus Single Variable

Question

Sketch the line whose Cartesian equation is given. $$ y-2 x=4 $$

Step-by-Step Solution

Answer

The line has a slope of 2 and crosses the y-axis at (0, 4).

Step 1: Understand the Line Equation

The given equation of the line is \( y - 2x = 4 \). This equation can be rewritten in the slope-intercept form \( y = mx + c \) for easier graphing.

Step 2: Rearrange the Equation

Rearrange the given equation to solve for \( y \) in terms of \( x \). Starting with \( y - 2x = 4 \), add \( 2x \) to both sides: \( y = 2x + 4 \). Now, the equation is in slope-intercept form, \( y = mx + c \), where \( m = 2 \) and \( c = 4 \).

Step 3: Identify the Slope and Y-Intercept

From the equation \( y = 2x + 4 \), the slope \( m = 2 \) and the y-intercept \( c = 4 \). This means the line crosses the y-axis at \( (0, 4) \) and has a slope of 2, which indicates it rises 2 units for every 1 unit it runs to the right.

Step 4: Plot the Y-Intercept

Begin by plotting the y-intercept \( (0, 4) \) on the graph. This is the point where the line intersects the y-axis.

Step 5: Use the Slope to Plot Another Point

Using the slope \( m = 2 \), which is \( \frac{2}{1} \), move 1 unit to the right (a positive x-direction) and 2 units up (a positive y-direction) from the y-intercept to locate another point on the line. Plot this point, \( (1, 6) \).

Step 6: Draw the Line

Once you have plotted at least two points \((0, 4)\) and \((1, 6)\), draw a straight line through these points. This line represents the graph of the equation \( y = 2x + 4 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way of expressing the equation of a line. It's written as \( y = mx + c \). This format is very user-friendly for graphing because it immediately shows you two important features: the slope \( m \) and the y-intercept \( c \). These two numbers help you easily plot the line on a graph.
When you rearrange a linear equation like \( y - 2x = 4 \) to the slope-intercept form, you simply solve for \( y \) in terms of \( x \). So, you add \( 2x \) to both sides: \( y = 2x + 4 \).

• \( m \) (slope) tells you how steep the line is.
• \( c \) (y-intercept) indicates the point where the line crosses the y-axis.

This form is preferred for graphing lines because these elements make it easy to see and draw the line.

Graphing Lines

When graphing lines, understanding the slope-intercept form makes the process much easier. First, locate the y-intercept on the graph—a point which in our example \( (0, 4) \) is where the line crosses the y-axis. This is always the first point to plot.
Next, use the slope to determine the direction and steepness of your line. From the y-intercept, move according to the slope. For a slope of 2, you'd move up 2 units for every 1 unit you move to the right.
When you plot these points, you’ll notice they fall in a straight pattern. Connect them to form a straight line.

• Starting at the y-intercept, use the slope to find more points.
• Draw a line through the points to graph as many as you need to fill the space.

This visual representation confirms the relationship the equation describes.

Slope

The slope \( m \) of a line measures its steepness and direction. It's often represented as a fraction, \( \frac{rise}{run} \), indicating how much the line rises vertically for each horizontal step. In the slope-intercept form \( y = mx + c \), \( m = 2 \) tells us the line rises 2 units for each 1 unit it moves to the right. This creates an upward slant from left to right.

• A positive slope means the line ascends from left to right.
• A negative slope would mean the line descends.
• A zero slope would make the line horizontal.

The slope is integral for plotting lines accurately and understanding their direction in a graph.

Y-Intercept

The y-intercept \( c \) is a key part of the slope-intercept form \( y = mx + c \). This is simply where the line crosses the y-axis. In other words, it’s the value of \( y \) when \( x = 0 \). For our equation, \( y = 2x + 4 \), the y-intercept is \( 4 \), which means the line hits the y-axis at the point \( (0, 4) \).
This intersection point serves as an anchor when plotting the line on a graph.

• Spot this point on the graph at the very start.
• Every other point on the line will be derived from this spot using the slope.

Knowing the y-intercept simplifies graphing and understanding real-world linear relationships, as it often represents the starting point before any changes occur.