Question

Write the equation in slope-intercept form. Then graph the equation. $$ y-0.5=0 $$

Short answer

The equation in slope-intercept form is \(y = 0.5\). The graph is a horizontal line crossing the y-axis at 0.5.

Step-by-step solution

Rewrite the equation in slope-intercept form

The given equation \(y - 0.5 = 0\) can be rewritten as \(y = 0.5\). This is already in slope-intercept form. The slope (\(m\)) is 0 (as no \(x\) term is present), and the y-intercept (\(c\)) is 0.5.

Graph the equation

Since the slope is 0, it means the line is horizontal. The y-intercept tells us that the line crosses the y-axis at 0.5. So, a horizontal line passing through the y-axis at 0.5 is graphed. It is parallel to x-axis.

Key concepts

Slope-Intercept Form

Graphing linear equations often starts with the slope-intercept form, a standard way to write linear equations. The slope-intercept form is expressed as
\( y = mx + b \),where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

Understanding the relationship between the slope and the y-intercept is fundamental in graphing equations. The slope indicates the steepness of the line and the direction in which it rises or falls. A positive slope means the line ascends from left to right, whereas a negative slope indicates it descends. When the slope is zero, the line is horizontal. No change in y-values implies no slope, as seen in the example equation \( y = 0.5 \).

In practical terms, after identifying \( m \) and \( b \) in an equation, you can quickly draw a rough sketch of the line. Start at the y-intercept on the y-axis, then use the slope to determine the next points on the line. In the example, since \( m \) is 0, there's no rise or run to calculate—simply draw a horizontal line across the graph at the y-intercept, \( b = 0.5 \).

Horizontal Line Equation

Horizontal lines run parallel to the x-axis and are characterized by an equation where the y-value is constant. For a horizontal line equation, the general form is \( y = k \),where \( k \) is the y-value constant across all points on the line. Since every point has the same y-coordinate, the slope of a horizontal line is always 0.

From the horizontal line equation \( y = 0.5 \) given in the exercise, it implies that no matter what x-value you choose, the y-value will always be 0.5. This creates a straight line that doesn't tilt up or down.

The graph of a horizontal line is straightforward to sketch. Simply find the constant y-value on the y-axis, in this case, 0.5, and draw a line straight across the entire graph. Because of their zero slope, horizontal lines indicate a situation where despite changes in one variable (x), the other variable (y) remains unchanged.

Y-Intercept

When it comes to graphing lines, identifying the y-intercept is crucial. The y-intercept is where the line crosses the y-axis, and it is represented by the \( b \) variable in the slope-intercept form of a line, \( y = mx + b \).

The y-intercept is an independent point where, regardless of the slope, the x-value is zero. It provides a starting position for the line on the graph and is especially useful when graphing a linear equation. You can apply this concept to the given exercise where the line crosses the y-axis at 0.5. This single point, \( (0, 0.5) \), is indispensable for plotting the horizontal line.

In real-world scenarios, the y-intercept can represent the initial value before any changes occur in association with the x-variable. For example, in economic models, the y-intercept might represent the startup cost of a business before any products are sold (the x-variable being the quantity of products sold). Understanding the y-intercept thus serves not only in mathematical graphing but also in interpreting data trends.