Problem 50 - Algebra 2

Question

OPEN ENDED Write an equation of a line with slope \(0 .\) Describe the graph of the equation.

Step-by-Step Solution

Answer

The equation is \( y = c \); it is a horizontal line parallel to the x-axis.

Step 1: Understand the Concept of a Horizontal Line

A line with a slope of zero is a horizontal line. The slope of a line indicates its steepness or inclination. A slope of zero means the line is perfectly flat and does not rise or fall as it moves from left to right.

Step 2: Identify the General Equation of a Horizontal Line

The general equation of a horizontal line is \[ y = c \]where \( c \) is a constant. The value of \( c \) is the constant \( y \)-coordinate for all points on the line, indicating that the line is horizontal and extends infinitely left and right at the height \( y = c \).

Step 3: Choose a Specific Value for \( c \)

To write a specific equation, choose a value for \( c \). For example, if \( c = 3 \), the equation will be \[ y = 3 \].This equation represents a horizontal line that crosses the \( y \)-axis at 3 and remains constant across the graph.

Step 4: Describe the Graph of the Equation

The graph of \( y = 3 \) is a horizontal line. It is a straight line parallel to the \( x \)-axis and passes through the point (0,3). Since the slope is zero, it does not slant and stays level across the entire graph.

Key Concepts

Slope of a Line

The concept of the slope is fundamental in understanding linear equations and their graphs. The slope is a measure of how steep a line is. It represents the rise or fall of the line as it moves from left to right.

• When the slope is zero, the line is horizontal, implying no vertical change as it travels along its length.
• Positive slopes indicate an upward trend, while negative slopes show a downward trend.
• The formula for slope is given by the ratio \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the y-coordinates, and \( \Delta x \) is the change in the x-coordinates.

When dealing with horizontal lines, the change in \( y \) is always zero because all points have the same \( y \)-value. Therefore, the slope \( m = \frac{0}{\Delta x} = 0 \). This means the line is flat and level.

Graphing Linear Equations

Graphing linear equations helps visualize solutions and understand line behavior on a plane. A linear equation describes a straight line, and its graph shows a continuous set of points.

• A common way to start graphing a line is by using the slope-intercept form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
• For horizontal lines, the equation simplifies to \( y = c \), where \( c \) is a constant representing the line's fixed height on the graph.
• Horizontal lines run parallel to the x-axis and intersect the y-axis at only one point, determined by the constant \( c \).

When producing these graphs, one can clearly see that, regardless of \( x \'s \) value, \( y \) remains constant, indicating the line's flat nature.

Constant Functions

Constant functions are straightforward yet important in mathematics. A constant function is a type of linear function where the output value is the same for any input value.

• The general form of a constant function is \( y = c \,\) where \( c \) is a fixed real number.
• Every x-value corresponds to the same \( y \)-value, so the graph is a horizontal line.
• Such functions represent situations where change does not occur, useful when modeling steady or unchanging conditions.

For instance, \( y = 3 \) is a constant function, indicating that at any point on the graph, the y-coordinate will always be 3. This demonstrates the concept of no variation with respect to x, aligning perfectly with the idea of a horizontal line.