Question

One of the following is an equation of the linear relation shown in the standard \((x, y)\) coordinate plane below. Which equation is it? A. \(y=5 x\) B. \(y=2 x\) C. \(y=5 x+2\) D. \(y=2 x-5\) E. \(y=2 x+5\)

Short answer

Options: A) y = 2x - 3 B) y = 2x + 3 C) y = -2x + 3 D) y = -2x - 3 E) y = 0.5x + 3 Answer: ___________

Step-by-step solution

Identify the slope and y-intercept

First, analyze the graph and identify the slope and the y-intercept of the linear relation. The slope of the line is the ratio of the vertical change to the horizontal change between two points on the line. The y-intercept is the point where the line crosses the y-axis.

Choose two points on the graph

Select two points on the graph of the linear relation. These two points can be any two distinct points on the line. If possible, choose points with integer values for their coordinates to simplify the calculation of the slope.

Calculate the slope

Using the two points selected in Step 2, calculate the slope of the line. The formula for calculating the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) are the coordinates of the first point and \((x_2, y_2)\) are the coordinates of the second point.

Find the y-intercept

Observe the graph and find the point at which the line intersects the y-axis. The y-intercept is the value of y when x = 0. Record the coordinates of this point.

Match the slope and y-intercept with given equations

Using the slope and y-intercept determined from the graph, compare them with the given options. The correct equation should have the matching slope and y-intercept. The selected equation from options A through E that matches the calculated slope and y-intercept will be the equation of the linear relation graphed on the standard \((x, y)\) coordinate plane.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is pivotal in graphing linear equations. It's defined as the equation of a line in the format of:
\[ y = mx + b \]
Here, \(m\) represents the slope of the line, which determines its steepness and direction. The slope is how much the line rises (or falls) vertically for each unit of horizontal movement to the right. Meanwhile, \(b\) is the y-intercept, which indicates where the line crosses the y-axis.
For instance, in the exercise given, the equations A to E are different variations of the slope-intercept form. By understanding how to read this form, you can quickly determine a line's characteristics just by looking at its equation. If the slope is positive, the line increases from left to right; if negative, it decreases. The y-intercept tells us the specific point on the y-axis where the line starts or crosses.
In simpler terms, this form allows you to graph a line by first plotting the y-intercept on the coordinate plane and then using the slope to find additional points on the line.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical values:
\( (x, y) \)
These values represent the horizontal position (x) and the vertical position (y) from the origin, which is the center of the plane where both x and y are zero. The plane is divided into four quadrants by the x-axis (horizontal line) and y-axis (vertical line).

Importance of the Coordinate Plane in Graphing

When graphing a linear equation, the coordinate plane allows you to visualize the relationship between the x and y values. By plotting points that satisfy the equation and connecting them, you create the graph of the line. In the exercise, you need to identify the correct graph, visualize it on the coordinate plane provided and then match it with the given equations. It's essential to know that the x-axis and y-axis are where lines often cross and these intersections are crucial for finding the slope and y-intercept.

Calculating Slope

The slope is a measure of how steep a line is on the coordinate plane. It is calculated by selecting any two distinct points on the line and using the formula:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]
A positive slope means the line ascends from left to right, while a negative slope indicates a descent. If the slope is zero, the line is horizontal and a vertical line has an undefined slope.
In the context of the exercise, choosing points with simple integer coordinates simplifies calculations. Calculate the slope using the provided points and ensure the consistency of the vertical and horizontal movements—'rise' refers to the change in y, and 'run' is the change in x. The consistent method of calculating the slope for various pairs of points confirms the line’s consistency in the graph.

Y-intercept Identification

Identifying the y-intercept is another essential step in understanding linear equations. It is the value of y in the equation when x is zero, representing the point where the line crosses the y-axis on the coordinate plane. To find the y-intercept from a graph, look at where the line meets the y-axis.
In the given exercise, after calculating the slope, you look at the graph to identify the y-intercept directly. It is where the vertical line running through the origin intersects the graph. If the graph doesn't exactly hit an integer value on the axis, you can estimate or read the value based on the scale of the graph. Then, simply compare the y-intercept to those found in the list of potential equations (A through E) to find the one that accurately represents the graphed line.