Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(0,5), m=-2.5$$

Step-by-Step Solution

Answer

The graph of the line with equation \( y = -2.5x + 5 \) passes through (0, 5) and (2, 0).

Step 1: Understand the Equation of a Line

The equation of a line can be given in slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We know the slope \( m = -2.5 \) and one point on the line \((0, 5)\), which is also the y-intercept \( b \). Thus, the equation of our line is \( y = -2.5x + 5 \).

Step 2: Find Another Point on the Line

To find a second point, we'll substitute a value for \( x \) into the line equation. Let's choose \( x = 2 \). Substitute into the equation: \( y = -2.5(2) + 5 \). Calculate: \( y = -5 + 5 = 0 \). So, the second point is \( (2, 0) \).

Step 3: Plot the Points

Now that we have two points: \((0, 5)\) and \((2, 0)\), we can plot them on a graph. The point \((0, 5)\), the y-intercept, is the starting point where the line crosses the y-axis. The second point \((2, 0)\) is located 2 units to the right and 5 units down.

Step 4: Draw the Line

With the two points \((0, 5)\) and \((2, 0)\) plotted, draw a straight line connecting them. This line extends infinitely in both directions and represents the equation \( y = -2.5x + 5 \).

Step 5: Label the Points

On the graph, make sure to label the points \((0, 5)\) and \((2, 0)\) for clarity. Indicating these points on your sketch helps anyone reading your graph understand where the line passes through.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It's expressed as \( y = mx + b \). Let's break this down:

• "\( y \)" represents the dependent variable, which is the output or the y-coordinate on a graph.
• "\( m \)" is the slope of the line. It tells us how steep the line is. A negative slope, like \(-2.5\) in our example, means the line will go down as it moves to the right.
• "\( x \)" is the independent variable, the input or x-coordinate.
• "\( b \)" is the y-intercept. It's the point where the line crosses the y-axis. In our exercise, \( b = 5 \), so the line crosses the y-axis at \((0, 5)\).

This form is especially user-friendly because it gives clear information about the slope and the starting point of the line.

Plotting Points

Plotting points is about putting specific points on a graph based on their coordinates. Each point is defined by an \((x, y)\) coordinate, which shows how far along the point is on both the x (horizontal) and y (vertical) axes.

• To plot \((0, 5)\), start from the origin (0,0) and move straight up 5 units. This is our y-intercept, meaning it's where our line will start.
• The second point \((2, 0)\) means move 2 units to the right from the origin, and since the y-coordinate is 0, you don't move up or down.

With these points marked, you can draw a line through them, representing the possible positions of \( y \), given all values of \( x \) along this line.

Equation of a Line

The equation of a line represents all the points that lie on the line. Each point satisfies the linear equation \( y = mx + b \). In our example:

• Our line's equation is \( y = -2.5x + 5 \), showing how each point on this line relates to \( x \) and \( y \).
• This specific equation indicates for every increase of 1 in \( x \), the \( y \) value decreases by 2.5 (because the slope \( m = -2.5 \)).
• Conversely, each decrease in \( x \) would increase \( y \) by 2.5. Notice that, by adjusting the x-value, you can find other points for the same line.

Understanding the equation of a line helps you predict or calculate specific points that fulfill the equation, making graphing straightforward and systematic.