Question

Give the equation and graph for a line with y-intercept and slope given in Exercises. $$y \text { -intercept }=2.5 ; \text { slope }=0$$

Short answer

Answer: The equation of the line is $$y = 2.5$$, and it represents a horizontal line on the graph passing through the y-intercept at (0, 2.5).

Step-by-step solution

Identify the slope and y-intercept

The problem provides us with the y-intercept and the slope: - y-intercept, b: 2.5 - Slope, m: 0

Write the equation of the line

Using the slope-intercept form, $$y = mx + b$$ and plugging in the given values (m=0 and b=2.5), we get the equation of the line as: $$y = 0x + 2.5$$ For simplification, the equation can be written as: $$y = 2.5$$

Graph the line

To graph the line $$y = 2.5$$, we can plot two points on the graph that satisfy the equation and connect them to form the line. Since the slope is 0, the line will be a horizontal line that passes through the y-intercept (2.5). 1. Fix x to 0, and find the corresponding y value: $$y = 2.5$$ Point A: (0, 2.5) 2. Fix x to 1, and find the corresponding y value: $$y = 2.5$$ Point B: (1, 2.5) Draw a horizontal line that passes through points A and B. This line represents the equation $$y = 2.5$$. In conclusion, the equation of the line is $$y = 2.5$$, and its graph is a horizontal line passing through the y-intercept at (0, 2.5).

Key concepts

Y-Intercept

The y-intercept is the point where a line crosses the y-axis of a graph. When you're given an equation of a line in slope-intercept form, which is written as y = mx + b, the y-intercept is represented by b. In practical terms, it tells you the value of y when x is zero.

Let's consider our current example, where the line provided has a y-intercept of 2.5. This means that the line will cross the y-axis at the point (0, 2.5). If you were to draw a vertical line through the graph at x = 0, the spot where it intersects our line is the y-intercept.

Understanding the role of the y-intercept is crucial as it provides a starting point for graphing the line and helps us visually interpret the equation.

Slope

The slope of a line is a measure of its steepness or the degree of its incline and is often represented by the letter m. Mathematically, slope is expressed as the ratio of the change in y (the vertical change) over the change in x (the horizontal change) between two distinct points on the line.

In our example, the slope is given as 0. This indicates that for every step we take to the right or left along the x-axis, there is no change in the y-coordinate. A slope of zero corresponds to a horizontal line – it does not rise or fall but remains constant, parallel to the x-axis.

A firm grasp of the concept of slope is essential, as it can convey the direction and the rate at which a line moves across the graph. Slopes can be positive, negative, zero, or undefined, each reflecting a different direction or type of line.

Slope-Intercept Form

The slope-intercept form is y = mx + b, where m represents the slope, and b represents the y-intercept of the line. This form is one of the most straightforward ways to write the equation of a line. It makes it easy to identify the slope and y-intercept, which are critical when graphing the line or finding points on it.

In our exercise, the slope-intercept form of the line was given as y = 0x + 2.5, which simplifies to y = 2.5, since any number multiplied by zero is zero. This simplified equation still fits within the slope-intercept structure. A slope of zero means the y-value remains constant no matter what x-value we choose. For students looking for clarity in working with linear equations, mastering the slope-intercept form is a valuable skill, as it immediately provides the key components needed to visualize a line.

Graphing Linear Equations

To graph linear equations, one must understand both the slope and the y-intercept, as these two values dictate how the line is drawn on a coordinate plane. The process generally involves plotting the y-intercept on the y-axis and then using the slope to determine another point on the line.

However, in the case of our horizontal line equation y = 2.5, graphing is straightforward. Since the slope is 0, we simply draw a horizontal line through the y-axis at the y-intercept point (0, 2.5). Whether you move left or right from this point along the x-axis, the y-value of the points on the line does not change. Therefore, the line extends left and right indefinitely at the height of y-value 2.5. This graphing method shows that even without a slope, one can still graph a line using just the y-intercept, as long as the slope provided is zero. Teaching students to graph linear equations enhances their ability to visualize algebraic expressions and solve various mathematical problems that involve linear functions.