Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=|4 x+3|+2 ;\) translation 2 units down

Short answer

The function \(g(x)\) which represents a translation of the function \(f(x)=|4 x+3|+2\), 2 units down, is \(g(x)=|4 x+3|\).

Step-by-step solution

Recall the Principle of Vertical Shifts

When we are shifting a graph vertically, we add or subtract an amount from the whole function. If the number is positive, the graph moves upwards; if negative, it moves downwards. Given that we are moving 2 units down, we subtract 2 from the function.

Apply the Principle

When we apply subtracting 2 to the function \(f(x)=|4 x+3|+2\), our function \(g(x)\) becomes \(g(x)=|4 x+3|+2 - 2\).

Simplify the Expression

On simplifying the expression we get \(g(x)=|4 x+3|\). This indicates that we have successfully moved the graph of \(f(x)=|4 x+3|+2\) 2 units down.

Key concepts

Understanding Vertical Shifts

When dealing with function transformations, vertical shifts are a fundamental concept. A vertical shift involves moving the entire graph of a function up or down along the y-axis. The rule of thumb is quite simple:

• To move a graph upwards, you add a positive value to the function.
• To move a graph downwards, you subtract a value from the function.

For instance, if the original function is denoted as \[ f(x) = |4x + 3| + 2 \] and we want to translate this function 2 units downwards, we subtract 2 from the entire function. This gives us the new function \[ g(x) = |4x + 3| + 2 - 2 \] which simplifies to \[ g(x) = |4x + 3| \]. This transformation effectively decreases all y-coordinates of the original graph by 2 units, visually shifting it down in the coordinate plane.

Exploring the Absolute Value Function

The absolute value function is another important piece of the function transformation puzzle. The absolute value of a number is its distance from zero on the number line regardless of direction, so it's always non-negative. The basic form of the absolute value function is \[ f(x) = |x| \].For the function we're discussing, \[ f(x) = |4x + 3| + 2 \],it is a stretched and shifted version of this basic function.

• The term \(4x + 3\) inside the absolute value modifies the input, stretching and shifting the graph horizontally.
• The "+2" allows the graph to move vertically, which we later adjusted by subtracting 2 for a downward shift.

The absolute value bars always ensure that the output is positive, which gives this function its characteristic "V" shape on a graph.

Using a Graphing Calculator

Graphing calculators are invaluable tools for visualizing functions and their transformations. They help us confirm our understanding and ensure accuracy in transformations like vertical shifts.Here’s how you can use a graphing calculator to check the transformation:

• Enter the original function, \( f(x) = |4x + 3| + 2 \), into the calculator.
• Next, input the transformed function, \( g(x) = |4x + 3| \), to see the effect of shifting 2 units down.
• Analyze the graph and compare both functions to visually confirm the downward shift.

This tool helps reinforce learning and enhances your ability to perform function transformations successfully, providing a visual confirmation of the algebraic changes made.