Problem 17
Question
For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x)-7 $$
Step-by-Step Solution
Verified Answer
The graph is shifted 7 units down.
1Step 1: Understand the Original Function
The provided function is a transformation of an original function. Let's assume the original function is \( y = f(x) \). We need to determine what effect the transformation has on this graph.
2Step 2: Identify the Transformation Type
The given function is \( y = f(x) - 7 \). The \'-7\' indicates a vertical shift. This transformation moves the graph of the function vertically downward.
3Step 3: Apply the Vertical Shift
Since \( y = f(x) - 7 \) involves subtracting 7 from the original function, every point on the graph of \( y = f(x) \) will move 7 units downwards. This affects the output value of the function corresponding to each x-value.
4Step 4: Describe the Resulting Graph
The resulting graph is exactly the same shape as \( y = f(x) \), but every point has been shifted straight down by 7 units on the y-axis. This does not affect the horizontal position at all.
Key Concepts
Understanding Vertical ShiftsWhat is a Function Graph?Defining the Original FunctionAll About Vertical Transformations
Understanding Vertical Shifts
Whenever we talk about a vertical shift, we are essentially referring to moving the entire graph of a function up or down along the y-axis. In the function transformation process, vertical shifts are one of the most straightforward to understand. To perform a vertical shift, you add or subtract a constant from the original function. If you add a positive constant, the graph moves up. Conversely, if you subtract a constant, like in the function \( y = f(x) - 7 \), the graph shifts downward.
In our example, each point on the graph simply moves straight down by 7 units as a result of the transformation \( y = f(x) - 7 \). This is because each y-value of the function is decreased by 7. It's important to note that vertical shifts only change the y-values and leave the x-values unchanged.
In our example, each point on the graph simply moves straight down by 7 units as a result of the transformation \( y = f(x) - 7 \). This is because each y-value of the function is decreased by 7. It's important to note that vertical shifts only change the y-values and leave the x-values unchanged.
What is a Function Graph?
A function graph is a visual representation of a function, showing the relationship between inputs (x-values) and their corresponding outputs (y-values). Graphs are powerful tools in mathematics because they allow us to visualize how a function behaves.
The graph of a function, such as \( y = f(x) \), is typically plotted on a two-dimensional plane with the x-axis (horizontal) and y-axis (vertical). Each point on the graph corresponds to an input-output pair \((x, f(x))\). By looking at the graph, we can easily see trends, patterns, and important features of the function, such as maxima, minima, and intervals of increase or decrease.
The graph of a function, such as \( y = f(x) \), is typically plotted on a two-dimensional plane with the x-axis (horizontal) and y-axis (vertical). Each point on the graph corresponds to an input-output pair \((x, f(x))\). By looking at the graph, we can easily see trends, patterns, and important features of the function, such as maxima, minima, and intervals of increase or decrease.
- The graph helps to visually understand transformations.
- It shows changes in position, shape, and orientation of a function.
Defining the Original Function
The original function is our starting point in this exploration of graph transformations. It is the basic form of the function before any transformations, such as shifts, stretches, or reflections, are applied. In mathematical notation, it's often presented as \( y = f(x) \).
Understanding the original function is critical because all transformations are relative to this initial form. For example, in the transformation \( y = f(x) - 7 \), we see how the graph of the original function \( y = f(x) \) is affected by the subtraction. Without knowing the original setup, it would be hard to appreciate how and why the graph changes.
Understanding the original function is critical because all transformations are relative to this initial form. For example, in the transformation \( y = f(x) - 7 \), we see how the graph of the original function \( y = f(x) \) is affected by the subtraction. Without knowing the original setup, it would be hard to appreciate how and why the graph changes.
- Transformation processes start with the original function.
- A clear understanding of \( y = f(x) \) allows for precise transformation application.
All About Vertical Transformations
Vertical transformations encompass changes that occur in the vertical direction on the function graph. They include vertical shifts, as well as vertical stretches and compressions. Each affects the graph of a function differently but similarly by altering the y-values of the function's output.
In vertical shifts, as we saw earlier with \( y = f(x) - 7 \), the graph moves up or down without changing its shape or orientation. In vertical stretches or compressions, the graph's shape is altered—either elongated or squished vertically. For example, multiplying the function by a constant greater than 1 stretches it, while a constant between 0 and 1 compresses it.
In vertical shifts, as we saw earlier with \( y = f(x) - 7 \), the graph moves up or down without changing its shape or orientation. In vertical stretches or compressions, the graph's shape is altered—either elongated or squished vertically. For example, multiplying the function by a constant greater than 1 stretches it, while a constant between 0 and 1 compresses it.
- Vertical shift moves the graph up or down.
- Vertical stretch/compression affects the graph’s height.
Other exercises in this chapter
Problem 16
For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{\sqrt{x+4}}{x-4} $$
View solution Problem 17
For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. $$ f(x)=\sqrt[3]{x-1} \text { and } g(x)=x^{3}
View solution Problem 17
For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph. $$ y=|x+1| $$
View solution Problem 17
Describe how the graph of the function is a transformation of the graph of the original function \(f.\) $$y=f(x)-7$$
View solution