Question

Your friend says two different translations of the graph of the parent linear function can result in the graph of \(f(x)=x-2\). Is your friend correct? Explain.

Short answer

No, the friend is not correct. Only one transformation of the parent function \( y = x \), a shift down by 2 units, is needed to get the function \( f(x) = x - 2 \).

Step-by-step solution

Understanding the Parent Function

Start off by understanding the light of the parent linear function. Here, the parent linear function is \( y = x \). The graph of this function is a line that passes through the origin (0,0) and rises at an angle of 45 degrees.

Identify the Given Function

The given function is \( f(x) = x - 2 \). This is a transformation of the parent function where every y-value is decreased by 2 units. This means that the line y=x is shifted 2 units downward.

Evaluate the Claim

The friend claims that two different translations of the parent function can result in the given function. However, only one transformation of the parent linear function is needed to get from \( y = x \) to \( f(x) = x - 2 \), which is shifting the function 2 units down. So the claim made by the friend is not correct.

Key concepts

Parent Linear Function

A parent linear function is essentially the simplest form of a linear equation and serves as the starting point for understanding transformations. In algebra, the parent linear function is given by the equation \( y = x \). The graph of this function is a straight line that bisects the first and third quadrants, passing through the origin with a slope of 1.

This means for every unit increase in the x-value, the y-value also increases by the same amount—thus, the 45-degree angle. It's crucial to nail down this foundation because all linear transformations will use it as their basis. Students should visualize this as a diagonal line on graph paper, where each square represents one step along both axes.

Graph Translations

Now, let's delve into graph translations. A translation is a type of transformation that shifts the graph horizontally and/or vertically without changing its shape or orientation. Think of it as sliding the entire graph to a new position.

For a linear function, translating a graph means we add or subtract a constant to the x or y values. For example, the function \( f(x) = x - 2 \) is the parent function \( y = x \) shifted 2 units downwards. Why downwards? Because we're subtracting from the y-value. Picture this as every point on the line being moved straight down by two 'steps'.

Contrary to some beliefs, multiple translations simultaneously (like sliding down and to the right) are considered one composite translation, not two separate actions.

Linear Function Transformations

Linear function transformations refer to various changes we can make to the parent function's graph. These can include translations, as mentioned earlier, but also stretches, compressions, and reflections.

For instance, by multiplying the x or y values, you change the slope or steepness of the line, resulting in a stretch (if larger than 1) or compression (if between 0 and 1). A negative multiplier would reflect the graph over the respective axis, flipping it.

Understanding these transformations is crucial because they help us visualize and predict the behavior of functions. It's like moving, stretching, or flipping a piece of string on a graph to model different scenarios.