Problem 45 - Algebra 2

Question

Graph the line with the given equation. \(y=2 x\)

Step-by-Step Solution

Answer

The line passes through the origin and rises 2 units for each unit it moves to the right.

Step 1: Understanding the Equation

The equation given is in slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this equation, \(y = 2x\), the slope \(m = 2\) and the y-intercept \(b = 0\). This means the line passes through the origin (0,0) and has a slope of 2.

Step 2: Plotting the Y-Intercept

Start by plotting the y-intercept on the graph. For the equation \(y = 2x\), the y-intercept is 0. So, place a point at the origin (0,0) on the graph.

Step 3: Using the Slope to Find Another Point

The slope of the line is 2, which can be written as \(\frac{2}{1}\). This means for every 1 unit you move to the right on the x-axis, move 2 units up on the y-axis. From the origin, move 1 unit to the right to x = 1, then 2 units up to y = 2, and plot this point (1,2).

Step 4: Drawing the Line

With the two points, (0,0) and (1,2), draw a straight line through both points, extending in both directions across the graph. This line represents the graph of the equation \(y = 2x\).

Step 5: Checking Additional Points

To verify, choose another point by using the slope or plugging a value into the equation to see if it lies on the line. For example, when \(x = 2\), \(y = 2 \times 2 = 4\). Plot the point (2,4) to confirm it's on the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations. This form is written as \( y = mx + b \), where \( m \) denotes the slope, and \( b \) is the y-intercept.
In this formula, the variables \( x \) and \( y \) represent any point on the line, but it's \( m \) and \( b \) that give us the tools to graph efficiently.

• The Slope (\( m \)): It tells us how steep the line is and the direction it goes, whether it's upwards or downwards.
  
• The Y-Intercept (\( b \)): This is where the line crosses the y-axis. It's the starting point for graphing our line.
  

To graph using the slope-intercept form, you start at the y-intercept (\( b \)) and use the slope (\( m \)) to find other points on the line. This method simplifies plotting linear equations on a graph.

Y-Intercept

The y-intercept of a line is a vital component because it indicates where the line will touch the y-axis. For any linear equation in the slope-intercept form \( y = mx + b \), the y-intercept is the value of \( b \).
In simpler terms, it is the point where \( x \) is zero.

To visualize the concept:

• Consider the point as "standing still" on the x-axis, as every change along x results in a different point on the line, except at the y-intercept.
• For instance, in the equation \( y = 2x \), the value of \( b \) is zero. Therefore, the line starts at the origin point (0,0) on the graph because that’s where \( y \) equals zero.
  

Understanding the y-intercept helps set the starting point of any line you need to graph.

Slope

The slope is all about direction and steepness. In mathematics, it plays a significant role in understanding how lines behave on a graph.
For an equation, written in slope-intercept form \( y = mx + b \), the slope is represented by the coefficient \( m \).


What the slope tells us:

• Rise over Run: The slope is calculated as \( \frac{{\text{rise}}}{{\text{run}}} \), meaning how many units the line goes up (rise) for a particular horizontal movement (run).
• Direction: A positive slope means the line ascends as you move from left to right; a negative slope means it descends.
  
• Steepness: A larger slope indicates a steeper line, while a smaller slope shows a flatter one.
  

For example, in the equation \( y = 2x \), the slope is \( 2 \). This means from any point on the line, moving 1 unit right on the x-axis, you move 2 units up on the y-axis. This rise over run helps pinpoint other points for accurate graphing.