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Magazine Chapter 3: Problem 24
For the following exercises, graph the given functions by hand. \(f(x)=-|x+3|+4\)
Short Answer
Expert verified
The graph of \(f(x)\) is a downward-opening V-shape with vertex at \((-3,4)\).
Step by step solution
01
Identify the Parent Function
The given function is based on the absolute value function, which is denoted by \(|x|\). In its basic form, the graph of \(y = |x|\) is a V-shaped graph with its vertex at the origin (0,0).
02
Identify Transformations
The function \(f(x) = -|x+3| + 4\) involves transformations of the parent function \(|x|\). The term \(x+3\) indicates a horizontal shift to the left by 3 units. The negative sign in front of the absolute value flips the graph upside down. The +4 adds a vertical shift upward by 4 units.
03
Plot the Vertex
After identifying the transformations, begin by plotting the vertex. For this function, the vertex occurs at the point \((-3,4)\) because of the horizontal shift left by 3 and the vertical shift up by 4.
04
Determine the Direction of the V-Shape
Because the function has a negative sign, the V-shape opens downward. This means the arms of the V will extend below the vertex point of \((-3,4)\).
05
Plot Additional Points
Choose additional points to the left and right of the vertex to help shape the graph. For easy calculations, select \(x = -4\) and \(x = -2\). For \(x = -4\), \(f(-4) = -|-4+3| + 4 = 3\). For \(x = -2\), \(f(-2) = -|-2+3| + 4 = 3\). Plot \((-4,3)\) and \((-2,3)\) on the graph.
06
Draw the Graph
Connect the vertex \((-3,4)\) and the additional points \((-4,3)\) and \((-2,3)\) with straight lines to form the V-shape that opens downward. Extend the lines beyond these points to fully depict the graph's shape.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Function
The absolute value function is a fundamental concept in mathematics. It is denoted as \(|x|\) and represents the distance of any number \(x\) from zero on the number line. Graphically, the absolute value function is characterized by its V-shaped graph. The basic absolute value function \(y = |x|\) has its vertex at the origin (0,0), where the two arms of the V meet.
- The arms of the graph extend outwards and upwards from the vertex.
- Each arm makes a 45-degree angle with the horizontal axis.
Function Transformations
Transformations help us modify the basic shape of a function's graph to fit specific functions. For the function \(f(x) = -|x+3| + 4\), we see several transformations from the basic \(|x|\) function.Transformations can include:
- Vertical shifts, which move the graph up or down.
- Horizontal shifts, which move the graph left or right.
- Reflections, which flip the graph across an axis.
Horizontal Shifts
Horizontal shifts change the left-right position of a function's graph. In the expression \(f(x) = -|x+3| + 4\), the inside of the absolute value, \(x+3\), indicates a horizontal shift. To determine the direction and magnitude of the shift, examine the expression:
- If the term is \(x-c\), the graph shifts right by \(c\) units.
- If the term is \(x+c\), the graph shifts left by \(c\) units.
Vertical Shifts
Vertical shifts alter the up-down position of a function's graph on a coordinate plane. In our function \(f(x) = -|x+3| + 4\), the part \(+4\) at the end causes a vertical shift.Here's how vertical shifts work:
- If the constant is positive, the graph moves up by that amount.
- If the constant is negative, the graph moves down by that amount.
