Problem 51
Question
\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift 4 units to the left and shift downward 2 units
Step-by-Step Solution
Verified Answer
The transformed graph's equation is \( f(x) = |x + 4| - 2 \).
1Step 1: Identify the Original Function
The original function is given as \( f(x) = |x| \). This represents the absolute value function, which is a V-shaped graph centered at the origin.
2Step 2: Apply Horizontal Shift
To shift the graph 4 units to the left, we replace \( x \) with \( x + 4 \) in the function. Thus, the transformed equation becomes \( f(x) = |x + 4| \).
3Step 3: Apply Vertical Shift
To shift the graph downward by 2 units, we subtract 2 from the entire function. So, the equation becomes \( f(x) = |x + 4| - 2 \). This represents a shift downward by 2 units.
Key Concepts
Absolute Value FunctionHorizontal ShiftVertical Shift
Absolute Value Function
The absolute value function, represented as \( f(x) = |x| \), is one of the fundamental building blocks in algebra. It forms a characteristic V-shaped graph that is symmetrical about the y-axis. This symmetry occurs because the function converts all negative inputs into their positive counterparts. As a result, any input value of \( x \), whether positive or negative, results in a positive output or zero.
- **Graph:** The graph of \( f(x) = |x| \) is structured such that it has a "vertex" at the origin (0,0). The graph rises equally in both directions from the vertex, giving it the V-shape.- **Applications:** This function is useful in real-world contexts where only the magnitude of a quantity is concerned, like distance or absolute magnitude.
Understanding how the absolute value function works is crucial for transforming it, which we'll cover in the next sections.
- **Graph:** The graph of \( f(x) = |x| \) is structured such that it has a "vertex" at the origin (0,0). The graph rises equally in both directions from the vertex, giving it the V-shape.- **Applications:** This function is useful in real-world contexts where only the magnitude of a quantity is concerned, like distance or absolute magnitude.
Understanding how the absolute value function works is crucial for transforming it, which we'll cover in the next sections.
Horizontal Shift
Horizontal shifts involve moving the entire graph of a function to the left or right on the x-axis. For the absolute value function \( f(x) = |x| \), a shift occurs by altering its equation. When you "shift 4 units to the left," you adjust the input variable in the absolute value function.
- **Mathematical Representation:** To achieve this shift, replace \( x \) with \( x + 4 \). This transforms the function into \( f(x) = |x + 4| \).- **Visual Change:** This alteration adjusts the vertex of the graph from (0,0) to (-4,0), effectively moving the entire graph four units to the left.
- **Why x+4?:** It may seem counterintuitive that adding 4 shifts the graph left. This is because you are modifying the input before the function's inherent properties apply.- **General Rule:** Remember, \( x - h \) shifts right by \( h \) units, \( x + h \) shifts left by \( h \) units.
This concept is pivotal for accurately moving specific features within function graphs.
- **Mathematical Representation:** To achieve this shift, replace \( x \) with \( x + 4 \). This transforms the function into \( f(x) = |x + 4| \).- **Visual Change:** This alteration adjusts the vertex of the graph from (0,0) to (-4,0), effectively moving the entire graph four units to the left.
- **Why x+4?:** It may seem counterintuitive that adding 4 shifts the graph left. This is because you are modifying the input before the function's inherent properties apply.- **General Rule:** Remember, \( x - h \) shifts right by \( h \) units, \( x + h \) shifts left by \( h \) units.
This concept is pivotal for accurately moving specific features within function graphs.
Vertical Shift
A vertical shift changes the position of a graph up or down along the y-axis. For our function \( f(x) = |x + 4| \), we'll apply this transformation by adjusting its output. This allows us to shift the graph "downward 2 units."
- **Mathematical Representation:** To implement the vertical shift, you subtract 2 from the entire function, resulting in \( f(x) = |x + 4| - 2 \).- **Visual Change:** This transformation moves the graph straight down by 2 units. The vertex, initially at (-4,0) after the horizontal shift, is now at (-4,-2).
- **Why Subtract 2?:** Subtracting lowers the output of the function, causing the downward shift.- **General Rule:** For a vertical shift: adding a constant moves it up; subtracting moves it down.
Mastering vertical shifts is essential for editing where functions sit on the graph and for meeting specific graphing requirements.
- **Mathematical Representation:** To implement the vertical shift, you subtract 2 from the entire function, resulting in \( f(x) = |x + 4| - 2 \).- **Visual Change:** This transformation moves the graph straight down by 2 units. The vertex, initially at (-4,0) after the horizontal shift, is now at (-4,-2).
- **Why Subtract 2?:** Subtracting lowers the output of the function, causing the downward shift.- **General Rule:** For a vertical shift: adding a constant moves it up; subtracting moves it down.
Mastering vertical shifts is essential for editing where functions sit on the graph and for meeting specific graphing requirements.
Other exercises in this chapter
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