Question

Graph the function and its parent function. Then describe the transformation. \(f(x)=x-6\)

Short answer

The given function \(f(x) = x - 6\) is a downshift of the parent function \(f(x) = x\) by 6 units.

Step-by-step solution

Graph the Parent Function

Begin with graphing the parent function, which is \(f(x) = x\). This is a straight line that passes through the origin with a slope of 1.

Graph the Given Function

Now, graph the given function \(f(x) = x - 6\). This is also a straight line. However, instead of passing through the origin like the parent function, it will intersect the y-axis at -6. The slope of this line also remains 1, since there is no transformation that affects the slope.

Describe the Transformation

Comparing the two graphs, it will be observed that the given function \(f(x) = x - 6\) is the result of shifting the parent function \(f(x) = x\) down by 6 units. This is because the subtraction of 6 in the given function translates to a downward shift in the graph. There is no stretching, compression, or reflection applied to the function, only a vertical shift.

Key concepts

Parent Function

At the core of understanding transformations is the concept of a 'parent function'. In essence, a parent function is the simplest form of a function family — it's the starting point. For linear functions, the most basic form is f(x) = x, which is a straight line passing through the origin (0,0) with a slope of 1. This simplicity makes it the ideal reference for observing transformations. When you grasp the idea of a parent function, you can better comprehend how various algebraic operations manipulate its graph into new shapes.

It's very similar to recognizing a family resemblance; just as you can spot common features in relatives, you can spot similarities between the transformed function and its parent function, regardless of the shifts or stretches applied.

Vertical Shift

A 'vertical shift' refers to the up or down movement of a graph of a function. When we have a function like f(x) = x - 6, it represents a vertical shift of the parent function f(x) = x. Specifically, the '-6' translates to moving the entire line down 6 units. If the number had been positive, the shift would have been upwards. Visualize this as picking up the graph line and simply sliding it along the y-axis, without any change in its slant or shape.

This concept is incredibly useful when dealing with real-world problems like adjusting projections for temperature changes over a week, calculating financial trends with a baseline shift, or even just understanding how a 'baseline' measurement can transform data.

Transformation of Functions

In mathematics, a 'transformation of functions' involves changing a function's position, shape, or size. This is done through various operations, such as shifting, stretching, compressing, or reflecting. When looking at the function f(x) = x - 6, the transformation is a vertical shift, and no other alterations are noted.

Transformation of functions is akin to morphing an object in a photo editing software — you can move it around, stretch it, flip it, or squash it. This concept is particularly valuable because it allows us to model and predict the behavior of complex systems in various scientific and engineering fields, simply by adjusting a standard set of function 'presets'.

Slope of a Line

The 'slope of a line' is a number indicating how steep the line is, mathematically representing the ratio of the vertical change ('rise') to the horizontal change ('run') between any two points on the line. For the parent function f(x) = x, the slope is 1, which indicates a 45-degree angle if we consider a unit circle. The slope of the transformed function f(x) = x - 6 remains unchanged. The line's steepness is the same; it's just been shifted down.

Understanding slope is not just academic; it's practical. It crops up in various real-world contexts, such as calculating the pitch of a roof, determining the grade of a road, or understanding economic growth rates. Recognizing the slope allows us to predict and compare the rates of change within these different scenarios.