Question

Graph the line corresponding to the equation \(y=2 x+1\) by graphing the points corresponding to \(x=0,1,\) and \(2 .\) Give the \(y\) -intercept and slope for the line.

Short answer

Answer: The slope of the line is \(2\), and the y-intercept is the point \((0, 1)\). Three points on the graph of this line are \((0, 1), (1, 3),\) and \((2, 5)\).

Step-by-step solution

Find the points on the graph

Substitute the given \(x\) values into the equation \(y = 2x + 1\) and solve for \(y\). For \(x = 0\): \(y = 2(0) + 1 = 0 + 1 = 1\) For \(x = 1\): \(y = 2(1) + 1 = 2 + 1 = 3\) For \(x = 2\): \(y = 2(2) + 1 = 4 + 1 = 5\) We now have three points on the graph: \((0, 1), (1, 3),\) and \((2, 5)\).

Plot the points on the graph

Plot the \((x, y)\) coordinates on a graph. Connect the points to create a straight line.

Determine the y-intercept

The \(y\)-intercept is the point where the line crosses the \(y\)-axis. When the line is in the form \(y = mx + b\), the y-intercept is represented as \((0, b)\). In this equation, \(y = 2x + 1\), and the y-intercept is the point \((0, 1)\).

Determine the slope

The slope of a line is represented by the coefficient of \(x\) when the equation is in the form \(y = mx + b\). In the given equation, \(y = 2x + 1\), the slope is \(2\). This means that for every increase of \(1\) in the \(x\) value, the \(y\) value increases by \(2\).

Key concepts

Y-Intercept

When you are graphing linear equations, the y-intercept is a fundamental concept to grasp. It's the spot where your line crosses the y-axis on a coordinate graph. Imagine the y-axis as a vertical line that goes up and down on your graph; the y-intercept is where your line will touch this axis.

In the equation of a straight line written as y = mx + b, the variable b represents the y-intercept. This is a number, not a pair of coordinates, which indicates at what point on the y-axis your line will pass through. For example, in the equation y = 2x + 1, the y-intercept is 1, meaning that the line will cross the y-axis at the point (0, 1).

Understanding the y-intercept is essential because it allows you to quickly plot the first point on a graph and gives you a starting place for drawing your line. It's particularly significant when solving and graphing real-world problems, as the y-intercept often represents a starting value or condition before changes begin to occur.

Slope of a Line

The slope of a line in mathematics describes how steep the line is. It is a measure of the line's incline, and it is crucial for understanding how the variables in an equation relate to one another.

A line's slope is commonly denoted by the letter m, and it represents the change in the vertical direction (y-value) for each unit of change in the horizontal direction (x-value). So in the equation y = mx + b, the coefficient of x (that is, m) is the slope. In your example of y = 2x + 1, the slope m is 2, indicating that for every one unit the x-value increases, the y-value will rise by two units.

The slope is critical for interpreting the rate of change between variables, such as speed, growth, or decrease. A positive slope means an overall increase, a negative slope points to a decrease, and a slope of zero indicates that there is no change. For those diving into calculus, this concept will evolve into the derivative of a function at a point.

Plotting Points on a Graph

Plotting points on a graph may seem straightforward, but it’s an essential skill for visualizing and solving linear equations. Each point on a graph has an (x, y) coordinate that corresponds to its horizontal and vertical positions.

To plot a point, begin by locating its x value on the horizontal x-axis, then move vertically to the y value. For example, if a point has coordinates (2, 3), you would move two units to the right along the x-axis and three units up along the y-axis. Where those two paths meet is where you place your point.

In the case of our linear equation y = 2x + 1, you would plot the points (0, 1), (1, 3), and (2, 5) following these steps. After you have plotted at least two points, you can draw a straight line through them to represent the linear equation. By practicing the plotting of points, you not only get better at creating graphs but also at interpreting them, an important skill for many scientific and analytical fields.