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Chapter 7: Problem 93

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(x)+6$$

Short Answer

Expert verified
The graph of function \(g(x)\) is a vertical shift upwards by 6 units from the graph of function \(f(x)\).

Step by step solution

01

Identify the transformation

Observe that the function \(g(x)\) is obtained by adding 6 to function \(f(x)\). This operation corresponds to a vertical shift in the graph. In specific, adding 6 means shifting the graph upwards by 6 units.
02

Visualizing the transformation

Try to visualize this on the graph. Every point \( (x, f(x)) \) on the graph of \(f(x)\) is moved to \( (x, f(x) + 6) \) on the graph of \(g(x)\), moving it 6 units upwards.
03

Summarize the relationship

The graph of \(g(x)\) is identical in shape to the graph of \(f(x)\), but it has been moved upward by 6 units. This is the relationship between the graph of \(g(x)\) and the graph of \(f(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Shift
In the world of function transformations, a vertical shift is quite straightforward. Simply put, it involves moving a graph up or down on the coordinate plane without altering its shape or orientation.
For instance, if we have a function like \[ g(x) = f(x) + c \] The graph will shift vertically by \(c\) units. A positive value of \(c\) will move it upwards, while a negative value will shift it downwards.
This means every point \((x, f(x))\) on the original graph will transform to \((x, f(x) + c)\) on the new graph.
  • Example: Moving from \(f(x)\) to \(g(x) = f(x) + 6\), each point goes up 6 units.
  • Consistency in Graph Shape: Despite the shift, the overall form of the graph remains unchanged.

This transformation is particularly helpful when adjusting graphs for different contextual situations in mathematics or real-world applications.
Graph Translation
Graph translation is about shifting a graph along the coordinate plane. When we talk about graph translation, it could involve movements both vertically and horizontally. However, in our specific example with \[ g(x) = f(x) + 6 \] we focus on vertical movement.
In this case, the translation only involves a vertical shift upwards by 6 units. The graph does not move left or right in any way.
Here鈥檚 how graph translation can generally be described:
  • Vertical translation occurs when a constant is added or subtracted from the entire function.
  • Horizontal translation affects the input variable and typically involves replacements like \(f(x+c)\) or \(f(x-c)\).

It's important to note how graph translation preserves shapes and only alters the graph's position, making it a simple yet powerful tool in algebra.
Algebra
Algebra is at the core of understanding transformations like vertical shifts and graph translations. It enables us to manipulate and interpret functions easily. In our focused example, the algebraic function transformation \[ g(x) = f(x) + 6 \] is a simple yet classic representation of algebra in action.
This illustrates how algebra is used to describe transformations that impact graphical representation.
Understanding these transformations through algebra helps us analyze and predict changes in a function鈥檚 graph through:
  • Simple operations such as addition and subtraction.
  • It provides a systemic way to visualize how each alteration affects the graph's position.

By utilizing basic algebraic expressions and rules, students can easily solve complex problems involving graph shifts and translations.

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