Question

Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Explain your reasoning. a. \(f(x)=2|x|-3\) b. \(f(x)=(x-8)^2\) c. \(f(x)=|x+2|+4\) d. \(f(x)=4 x^2\)

Short answer

a. Vertical translation, b. Horizontal translation, c. Both horizontal and vertical translations, d. Neither horizontal nor vertical translations.

Step-by-step solution

Identify Parent Functions and Translations

The parent function of \(f(x)=2|x|-3\) is \(|x|\). Comparing the two functions, it's apparent that there is a vertical translation (down by 3 units), because of the -3. However, there is no horizontal translation since there is no value added or subtracted inside the absolute value.

Identify Parent Functions and Translations

The parent function of \(f(x)=(x-8)^2\) is \(x^2\). Comparing the two functions, it's apparent that there's a horizontal translation (to the right by 8 units), because of (x-8). However, there is no vertical translation since there is no value added or subtracted outside the parenthesis.

Identify Parent Functions and Translations

The parent function of \(f(x)=|x+2|+4\) is \(|x|\). Comparing the two functions, it's apparent that there's a horizontal translation (to the left by 2 units, because of the (x+2)) and a vertical translation (up by 4 units), because of the +4.

Identify Parent Functions and Translations

The parent function of \(f(x)=4x^2\) is \(x^2\). Comparing the two functions, there is no horizontal or vertical translation, as the function is just a scaling of the parent function by a factor of 4, which does not affect the position of the graph, but its shape.

Key concepts

Horizontal Translation

A horizontal translation moves the graph of a function left or right along the x-axis.
This occurs when we add or subtract a constant value inside the function’s expression that involves the variable x.
Imagine sliding a picture horizontally without changing its size or shape.For example, consider the function \(f(x) = (x - 8)^2\). This function is derived from its parent function \(x^2\).
The expression \((x - 8)\) indicates a horizontal translation to the right by 8 units.
This is because subtracting a number moves it to the right, whereas adding would move it to the left.Horizontal translations are powerful because they help adjust the position of a function's graph without modifying its overall appearance. This is incredibly useful when modeling shifts in real-life scenarios.

Vertical Translation

A vertical translation shifts the graph up or down along the y-axis.
This happens by adding or subtracting a constant outside the function's main formula or expression.
It's like lifting or lowering a picture without altering its horizontal position.Take the function \(f(x) = 2|x| - 3\). The parent function here is \(|x|\).
We see that the '-3' causes the entire graph to move downward by 3 units.
Unlike horizontal translations, vertical translations affect how high or low the graph appears.Vertical translations can be helpful in practically elevating or lowering positions in graphs, such as adjusting for different baselines in data modeling.

Parent Functions

Parent functions are the simplest form of functions from which other complex functions are derived.
They provide a basic template that helps understand and make connections with various transformations like translations or scaling.For instance, the parent function of \(f(x) = 4x^2\) is \(x^2\).
The function \(x^2\) is a simple polynomial function whose graph is a parabola opening upward.
Parent functions act as a benchmark to see the effect of transformations applied to the function.Knowing parent functions helps in predicting the shape and behavior of its transformed offspring, enabling us to describe and analyze complex models with relative ease.

Absolute Value Functions

Absolute value functions are characterized by the notation \(|x|\), which represents the distance of x from zero.
They are inherently non-negative, meaning they always output values greater than or equal to zero.The parent function of an absolute value expression is \(f(x) = |x|\).
For instance, in the function \(f(x)=|x+2|+4\), the parent function is \(|x|\).
Here, an absolute value creates a V-shaped graph that is centered on the y-axis when the parent is unmodified.These functions are often used in scenarios that require measuring distance regardless of direction, or in piecewise function that reflect symmetry around a point.
By understanding absolute value functions, students gain insight into nonlinear systems and how they can be manipulated with various translations or transformations.