Question

Write the slope-intercept form of the equation of the line that passes through the two points. Graph the line. Label the points where the line crosses the axes. \((-1,4),(2,4)\)

Short answer

The slope-intercept form of the line that passes through the points (-1, 4) and (2, 4) is \(y = 4\). When graphed, this line is a straight horizontal line passing through y = 4 on the y-axis. The line doesn't cross the x-axis.

Step-by-step solution

Compute the Slope (m)

The slope formula is given as \(m = (y2 - y1) / (x2 - x1)\) where (x1, y1) and (x2, y2) are coordinates of two points. The provided points are (-1,4) and (2,4). Plug into the formula to get: \(m = (4 - 4) / (2 - (-1)) = 0 / 3 = 0\)

Substitute a Point and the Slope into the Equation

One can choose either of the points for this step. Here (-1, 4) is used. So, substitute these into the slope-intercept formula to get: \(4 = 0*(-1) + c\). Solving for 'c' gives: c = 4.

Write the Equation of Line

Substitute 'm' and 'c' into the slope-intercept form: y = mx + c. This gives: \(y = 0x + 4\) or simplified to \(y = 4\).

Graph the Line

On the graph, plot the line that passes through the points (-1,4) and (2,4), which is a horizontal line that intersects the y-axis at y = 4. Label this point. The line does not intersect the x-axis.

Key concepts

Coordinate Graphing

Coordinate graphing is a method of representing points in a plane using a pair of numbers. These numbers, or coordinates, define the position of a point in relation to two intersecting lines known as the x-axis (horizontal) and y-axis (vertical). The meeting point of these two axes is called the origin, labeled as (0,0).

When you plot a pair of coordinates, such as (-1, 4), the first number indicates the position along the x-axis, and the second number shows the position along the y-axis. In coordinate graphing, it's crucial to understand that the order of these numbers matters, as it defines the exact placement of the point on the graph.

A line on a coordinate plane can be determined by plotting at least two points and drawing a straight line through them. This process is integral to graphing equations and understanding how they visually represent relationships between variables.

Writing Equations of Lines

Writing equations of lines is an essential skill in algebra. An equation of a line provides the relationship between the x and y coordinates of any point that lies on the line. The slope-intercept form is one of the most common ways to write the equation of a line, represented as: \( y = mx + c \) where 'm' is the slope of the line, and 'c' is the y-intercept.

Slope-intercept form is particularly useful because it clearly shows both the slope and where the line crosses the y-axis. To write an equation in this form, you need to know both of these values. As in the example above, with the coordinates (-1, 4) and (2, 4), the slope is 0, which indicates a horizontal line, and the y-intercept is 4, which shows where the line crosses the y-axis.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Mathematically, it is calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line, often described as 'rise over run.'

The formula for slope is: \( m = \frac{y2 - y1}{x2 - x1} \) where \( (x1, y1) \) and \( (x2, y2) \) represent the coordinates of two points on the line. If the slope is positive, the line rises as it moves from left to right. Conversely, a negative slope indicates that the line falls. A slope of zero, as in the exercise, describes a horizontal line, and an undefined slope (which occurs when the denominator is zero) corresponds to a vertical line.

Understanding the slope is crucial for graphing the line and predicting the behavior of the line as it extends in both directions on the coordinate plane.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a coordinate graph. It is represented by the 'c' in the slope-intercept equation \( y = mx + c \). This value gives us direct insight into where the line will touch the y-axis when the value of x is zero.

Identifying the y-intercept is a fundamental step when graphing a line or writing its equation in slope-intercept form. As seen in the example, solving for 'c' involved substituting the slope and the coordinates of a point on the line into the slope-intercept equation and solving for y. The resultant y-intercept was 4, indicating that the line intercepts the y-axis at the point (0, 4).

It's important to note that every straight line will have a single y-intercept, which simplifies the process of graphing and understanding linear equations. This is one of the characteristics of linear functions that makes them readily analyzable and applicable to a variety of problems in mathematics and applied disciplines.