Problem 41 - Thinking Mathematically

Question

In Exercises 41-48, graph each horizontal or vertical line. \(y=4\)

Step-by-Step Solution

Answer

The graphical representation of the equation 'y = 4' is a horizontal line that intersects the y-axis at point 4.

Step 1: Understand the given equation.

Examine the equation which is 'y = 4'. This is a straightforward linear equation in which y is isolated. In such equations, the value of y is always constant and it is determined by the number on the right side of the equation. In this case, it is 4.

Step 2: Interpreting the equation graphically

In the equation 'y = 4', there is no 'x' variable involved. This suggests that the line doesn't slope but it is rather horizontal. The number indicates the position of the line in relation to the y-axis. Since it is 4 in this case, the line will sit on the position 4 on the y-axis.

Step 3: Plot the line on the graph

Take a graph paper or use a graphing tool of your choice. On the y-axis, find the position that corresponds to the number 4. Draw a horizontal line across the graph at this point. This is the line which the equation 'y = 4' represents.

Key Concepts

Horizontal Lines

Horizontal lines are one of the simplest types of lines in the coordinate plane. They run straight across from left to right, parallel to the x-axis. But what makes a line horizontal? It's all about the y-value.

• For any horizontal line, the y-coordinate remains constant across all its points.
• The equation for a horizontal line is in the form of: \(y = c\), where \(c\) is a constant value.
• This means no matter which point you choose on the line, the y-value will always be the same.

To visualize this on a graph, just think of a horizon—an even, flat line that doesn't go up or down. The line will pass through the y-axis at the given constant y-value and extend infinitely in both directions.

Coordinate Plane

The coordinate plane is a fundamental concept in graphing linear equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.

• The intersection point of the x-axis and y-axis is called the origin, represented by the coordinates (0, 0).
• The x-values increase to the right of the origin and decrease to the left.
• The y-values increase as you move up from the origin and decrease as you move down.

When graphing an equation, points are plotted according to their x and y values, creating a visual representation. For the line \(y = 4\), you would move 4 units up from the origin on the y-axis and draw a line parallel to the x-axis without changing the y position.

Equations of Lines

Equations of lines are mathematical expressions that describe the relation between the x-values and y-values of any point on the line. They can be simple or complex depending on the type of line described.

• The most common form of the equation of a line is the slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Horizontal lines are a special case where the slope \(m = 0\), so the equation simplifies to \(y = b\).
• In these cases, as we saw with \(y = 4\), the y-value is constant and solely depends on \(b\), the y-intercept.

Understanding these equations helps in easily identifying the nature of the line, such as its slope and where it crosses the y-axis. For horizontal lines, it’s all about recognizing that the slope is zero, which simplifies interpretation and graphing.