Problem 1 - Precalculus Mathematics for Calculus

Question

Fill in the blank with the appropriate direction (left, right, up, or down). (a) The graph of \(y=f(x)+3\) is obtained from the graph of \(y=f(x)\) by shifting _____ 3 units. (b) The graph of \(y=f(x+3)\) is obtained from the graph of \(y=f(x)\) by shifting _____ 3 units.

Step-by-Step Solution

Answer

(a) up 3 units, (b) left 3 units.

Step 1: Analyze Expression for (a)

The expression given is \(y = f(x) + 3\). This represents a vertical shift of the graph of \(y=f(x)\). When we add a constant to the function, it shifts the graph upwards by that constant.

Step 2: Determine Shift for (a)

Since the expression involves adding 3 to the function, the graph undergoes a vertical shift. Specifically, the graph shifts **up** 3 units.

Step 3: Analyze Expression for (b)

The expression given is \(y = f(x+3)\). This represents a horizontal shift of the graph of \(y=f(x)\). When a positive constant is added inside the function's argument, the graph shifts in the opposite direction on the x-axis.

Step 4: Determine Shift for (b)

Adding 3 inside the argument of the function shifts the graph horizontally. In this case, the graph shifts **left** 3 units.

Key Concepts

Vertical Shift

A vertical shift in a graph occurs when a constant is added or subtracted from the function itself. To picture this, imagine wrapping the entire graph of a function and moving it straight up or down, depending on whether the constant is positive or negative.
When you add a constant to the entire function, as in the expression \(y = f(x) + 3\), the graph will shift upward by that constant. Here's why:

• Every point on the graph moves up by exactly the same number of units as the constant added.
• Vertical shifts do not affect the shape of the graph; they only change its vertical position.

This means that if you see \(y = f(x) + 3\), you know the graph has gone up 3 units, moving the entire graph parallel to its original path along the y-axis. Vertical shifts are straightforward: add to go up and subtract to go down.

Horizontal Shift

Horizontal shifts involve moving a graph left or right along the x-axis. This happens when we add or subtract a constant inside the function's argument, meaning it affects the input values directly. A common pitfall is that these shifts work in the opposite direction of what you might intuitively think.
Consider the expression \(y = f(x + 3)\). Here, you add 3 to the input \(x\), which sounds like you want to move right, but it actually moves the graph to the left. Let's understand why:

• Adding a constant inside the function argument \((x + 3)\) pushes the graph in the opposite direction, meaning left.
• Subtracting a constant would move the graph right.
• Horizontal shifts, like vertical ones, do not change the shape of the graph.

So, for \(y = f(x+3)\), the graph shifts left 3 units. It's key to remember that adding inside equals left, and subtracting equals right.

Function Translation

Function translation encompasses both vertical and horizontal shifts, allowing a function's graph to move around the coordinate plane without rotation or changing its shape. By utilizing both types of shifts, a graph can be strategically positioned.
Here's how function translation works together:

• Vertical shifts move the function up or down on the y-axis depending on whether a constant is added or subtracted from the function output \(f(x)\).
• Horizontal shifts move the graph left or right along the x-axis when the constant is added or subtracted inside the argument \(x\).
• The overall graph position is determined by the sequence and values of these shifts.

This translation is useful for matching real-world scenarios to mathematical models, as it allows for easy adjustments to fit data points. In combining both vertical and horizontal shifts, any function can be translated to any location on the graph, maintaining its shape in the process.