Problem 89 - Elementary Algebra

Question

Graph. $$ y=(x-1) 2 $$

Step-by-Step Solution

Answer

The graph of \( y = 2(x - 1) \) is a straight line with a slope of 2 and a y-intercept of -2.

Step 1: Understand the equation

The given equation is a linear equation of the form \( y = m(x - 1) \). To match our standard linear equation form \( y = mx + b \), recognize that \( y = 2(x-1) \). It describes a line with a slope of 2, but we should expand it to easily identify the slope and y-intercept.

Step 2: Expand the equation

To understand where the line will be on the graph, expand the equation to make it into the standard linear form. Starting with the equation \( y = 2(x-1) \), distribute the 2: \( y = 2x - 2 \). This means the slope (m) is 2, and the y-intercept (b) is -2.

Step 3: Determine key points

Identify key points to plot the line. Use the y-intercept: when \( x = 0 \), \( y = -2 \). This gives the point (0, -2). The slope is 2, which means for every 1 unit increase in \( x \), \( y \) increases by 2 units. From (0, -2), moving 1 unit to the right along the x-axis (x=1), would increase \( y \) by 2 to \( y = 0 \), giving the point (1, 0).

Step 4: Plot the points and graph the line

On a graph, plot the points (0, -2) and (1, 0). Draw a straight line through these points, extending across the axes. This is the graphical representation of the equation \( y = 2(x - 1) \).

Step 5: Verify the graph

To ensure accuracy, choose another value for \( x \) and check if the graph corresponds. For example, if \( x = 2 \), then \( y = 2(2-1) = 2 \). The point (2, 2) should be on the line. Verify this on the graph.

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a common way to express linear equations. It is written as \( y = mx + b \), where:

• \( m \) is the slope, which measures the steepness or incline of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In our exercise, the equation starts as \( y = 2(x-1) \), which isn't in the standard slope-intercept form yet.
To put it into this form, we need to distribute the 2, resulting in \( y = 2x - 2 \).
Now, it's clear: the slope \( m \) is 2, and the y-intercept \( b \) is -2.
Understanding the slope-intercept form helps us by making it easier to graph the line,
since it gives us the slope and y-intercept directly.

Plotting Points to Graph Lines

Plotting points is a crucial part of graphing linear equations. It's about marking specific points on the graph,
which help us draw the line accurately. With our equation of \( y = 2x - 2 \) already in slope-intercept form, we start by identifying key points.
Start with the y-intercept, \( b = -2 \).
This is the point (0, -2), which tells us the line crosses the y-axis here.
Next, use the slope \( m = 2 \), indicating that for every 1 unit right on the x-axis, y increases by 2 units.
From (0, -2), moving right to x = 1, y becomes 0, giving us point (1, 0).
By plotting these two points, we have enough information to draw a straight line.
This line shows the graph of the equation and can be extended in both directions.
To ensure our line is accurate, check another point like x = 2, where y should equal 2.
This should be on the line as well, verifying correctness.

The Nature of Linear Functions

Linear functions are a type of function represented by lines on a graph.
They always have a constant rate of change, meaning their slopes are fixed.
This means every increase in x corresponds to a proportional increase or decrease in y.
The equation form \( y = mx + b \) gives us insight into these characteristics.
In the context of our example, the function \( y = 2x - 2 \) shows a linear relationship,
where doubling x doubles y, due to the slope being 2.
This demonstrates the consistent rate of change typical of linear functions.
Understanding this helps in predicting y for any given x.
Hence, linear functions are fundamental in mathematics as they simplify complex relationships,
turning them into straightforward, predictable patterns easily represented on graphs.