Problem 35 - A Graphical Approach to College Algebra

Question

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=-0.75 x$$

Step-by-Step Solution

Answer

Plot (0,0) and (4,-3), then draw a line through them.

Step 1: Understanding the Equation

The equation given is \(y = -0.75x \). This is a straight line passing through the origin \((0,0)\) with a slope \(k = -0.75\). The negative slope indicates the line will slant downwards.

Step 2: Calculate the Second Point

To find another point on the line, pick a simple value for \(x\). Choose \(x = 4\), then plug it back into the equation: \(y = -0.75 \times 4 = -3\). So, the second point is \((4, -3)\).

Step 3: Plot Points on Graph

Plot the origin \((0,0)\) and the point \((4, -3)\) on the coordinate plane. Mark these two points clearly.

Step 4: Draw the Line

Using a ruler, draw a straight line that passes through both points \((0,0)\) and \((4, -3)\). This line represents the equation \(y = -0.75x\).

Key Concepts

Slope-intercept form

When dealing with linear equations like the one in the exercise, you may encounter the slope-intercept form. This is a common way to express the equation of a line, usually given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. In the simplified version presented in the original exercise, the equation \( y = -0.75x \) indicates that the y-intercept is 0. This is because the line passes through the origin, which means \( b = 0 \).

• The slope, in this case, is \( -0.75 \).
• A negative slope means the line goes downward as it moves from left to right on the graph.

Understanding these components helps us to visualize and graph such linear equations effectively.

Coordinate plane

The coordinate plane is a two-dimensional surface on which we can graph lines and curves. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. Each point on the coordinate plane is identified by an ordered pair \( (x, y) \), indicating its position with respect to the origin \( (0, 0) \).

• The origin is the point where both axes intersect.
• Quadrant I is where both x and y are positive.
• Quadrant II is where x is negative and y is positive.
• Quadrant III is where both x and y are negative.
• Quadrant IV is where x is positive and y is negative.

For our equation \( y = -0.75x \), it passes through the origin and uses the coordinate plane to depict the line’s direction and slope.

Plotting points

Plotting points is one of the fundamental steps in graphing equations. It involves identifying specific points on the coordinate plane that lie on the graph of an equation. To plot a point, write down its coordinates as \( (x, y) \). Then, start from the origin. Move horizontally to the \( x \)-coordinate and then vertically to the \( y \)-coordinate. Mark the spot where this leads you.In our exercise, we first plot the origin \( (0, 0) \). Then, we choose another point to ensure accuracy when drawing the line. We found the second point to be \( (4, -3) \) by substituting \( x = 4 \) into the equation and finding that \( y = -3 \).

• Plotting the origin \( (0,0) \) confirms the starting point since our line passes through it.
• The point \( (4, -3) \) confirms the slope and direction of the line.

With these two points marked, a straight line can be drawn to represent the equation, ensuring it aligns with the slope provided.