Question

Graph the function and its parent function. Then describe the transformation. \(f(x)=3\)

Short answer

The parent function is \(f(x)=0\). The given function, \(f(x)=3\), is a transformation of the parent function where it is shifted vertically upwards by 3 units.

Step-by-step solution

Identify the Parent Function

The parent function of \(f(x)=3\) is \(f(x)=0\). This is because \(f(x)=3\) is a vertical shift of the x-axis, which can be represented by \(f(x)=0\).

Graph the Parent Function

Plot the line \(f(x)=0\), which is the x-axis. It's a straight line that passes through the origin (0,0) and extends indefinitely in both the positive and negative direction.

Graph The Given Function

Now, plot the line \(f(x)=3\). This is also a straight line that is parallel to the x-axis, but instead of passing through the origin, it passes through the point (0,3) because the entire function, no matter the value of x, always equals 3.

Describe the Transformation

The transformation that turns the parent function \(f(x)=0\) into the given function \(f(x)=3\) is a vertical shift upwards by 3 units. This is because for every x-value, the y-value of the function has increased by 3 as compared to the parent function.

Key concepts

Parent Function

A parent function is the simplest form of a set of functions that form a family. In essence, it serves as the basic blueprint from which other functions in the same family can be derived through various transformations. Think of it as a genetic ancestor for functions! For linear functions, like the one dealt with in our exercise, the parent function is typically represented by f(x) = 0. It’s a horizontal line that crosses the y-axis at the origin. This simplicity makes it an excellent starting point for understanding more intricate function behaviors, as any transformation applied to this baseline is clearly observable.

Vertical Shift

A vertical shift is one of the most straightforward types of function transformations. It moves a graph up or down without changing its shape or orientation. You can easily perform a vertical shift by adding or subtracting a constant to the parent function's output (y-value). In the provided exercise, we can see a vertical shift executed by adding 3 to the parent function, resulting in f(x) = 3. Each point on the parent function’s graph is moved up by 3 units, which is visible in the final graph as a parallel line to the original, now passing through the point (0, 3).

Function Transformation

Function transformation is a broad term that encompasses all the operations that can alter the appearance of a graph. This includes vertical and horizontal shifts, stretching and shrinking, reflections, and rotations. Transformation takes a parent function and systematically churns out a related function with a new position or size. Understanding the base parent function is key, as transformations apply a set of algebraic rules to it. For instance, when we apply a vertical shift, as in our exercise, we're using a rule that modifies the y-value for every x-value uniformly, effectively translating the graph along the y-axis.

Plotting Functions

Plotting functions is a fundamental skill in graphing transformations. It involves drawing the graph of a function by marking points that satisfy the function's equation on a coordinate plane. The more points you plot, the more accurately the curve or line can be drawn. In our exercise, plotting starts with the parent function and with the given function. By plotting and visually comparing both, students gain a clear understanding of the impact transformations have on graphs. Techniques for plotting can include creating a table of values or applying known transformations to points from the parent function's graph.

For exercises like these, one would plot the parent function first, which lays a foundation for comparison. Next, by plotting the transformed function, in this case by simply shifting all points up by 3 units, one can easily observe the vertical shift transformation.