Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Explain how the graph of \(g(x)=f(x+4)\) is obtained from the graph of \(y=f(x)\).

Step-by-Step Solution

Answer

The graph is shifted 4 units to the left.

Step 1: Understanding Function Transformation

To understand how the graph transforms, we need to look at the change in the input of the function. The original function given is \(y = f(x)\). The new function is \(g(x) = f(x+4)\). Here, the input \(x\) is replaced with \(x+4\).

Step 2: Analyzing the Transformation Effect

Replacing \(x\) with \(x + 4\) inside the function causes a horizontal shift in the graph. This is because the input is being adjusted inside the function, affecting how the graph appears horizontally.

Step 3: Identifying the Type of Shift

When you modify the input \(x\) to \(x + 4\), it means every point on the graph of \(y = f(x)\) will move to the left by 4 units. This is a horizontal shift to the left.

Step 4: Visualizing the Transformation

Graphically, if you were to plot the graph of \(y = f(x)\), then the graph of \(g(x) = f(x+4)\) would look exactly the same but every point would appear shifted 4 units to the left compared to \(f(x)\).

Key Concepts

horizontal shift

A horizontal shift involves moving the entire graph of a function left or right along the x-axis. Imagine you have a graph on squared paper. A horizontal shift will move the picture of the graph to a new position left or right without changing its shape.

• When we add a positive number inside the function (e.g., replacing \(x\) with \(x+4\)), the graph shifts to the left.
• If we subtract a number (e.g., \(x-4\)), the graph shifts to the right.

This shifting occurs due to changes in the input values, which are governed by the function's equation.

The main idea is that adjusting the input directly alters where the entire graph sits on the x-axis without affecting the vertical component.

function graph

The function graph is a visual representation of the relationships defined by a function. Think of it as the picture that forms when we plot all the possible pairs of input and output values from a function on a set of axes (often x and y).

• The x-axis usually represents the input values.
• The y-axis represents the corresponding output values.

When seeing a graph, the shape and position show us how the outputs change with different inputs. A function graph can reveal important characteristics, like:

• Where the function increases or decreases
• The function's symmetry
• Intercepts with axes

Understanding these features helps us analyze transformations like shifts and stretches which alter the graph's position or shape in a defined way.

input modification

Input modification is all about changing the part of a function where we insert values—usually replacing \(x\) with something new that contains \(x\), like \(x+4\). This can change how inputs map to outputs, directly affecting the graph's appearance or position.

For the transformation \(g(x) = f(x+4)\), input modification causes each x-value to be recognized 4 units earlier than it normally would be. So, each input value shifts effectively moving the graph left.

Understanding input modifications can help determine:

• How shifts along the axes occur.
• Why certain graph positions or orientations change.

By thoroughly grasping input modifications, we can predict the resulting transformations in a graph and how these tweaks impact its overall structure.