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Magazine Chapter 2: Problem 8
Sketch a graph of each function $$ f(x)=3|x-2|-3 $$
Short Answer
Expert verified
The graph is a 'V' shape with vertex at (2,-3), opening upwards, stretched vertically by 3.
Step by step solution
01
Identify the Basic Function
The given function is based on the absolute value function, \( f(x) = |x| \). The absolute value function forms a 'V' shape with a vertex at the origin (0,0).
02
Identify Transformations
The function \( f(x) = 3|x-2|-3 \) involves several transformations from the basic absolute value function.1. Horizontal Shift: \( |x-2| \) indicates a shift of the graph 2 units to the right.2. Vertical Stretch: The coefficient 3 indicates a vertical stretch by a factor of 3.3. Vertical Shift: The \(-3\) at the end indicates a downward shift by 3 units.
03
Determine the Vertex
The transformations provide the new vertex of the graph. Start from \((0,0)\) and apply the shifts:- First, move 2 units to the right: \((0+2, 0)\) becomes \((2,0)\).- Then move down 3 units: \((2,0-3)\) becomes \((2,-3)\).Thus, the vertex of the graph is at \((2, -3)\).
04
Sketch the Graph
1. Plot the vertex at \((2,-3)\) on the coordinate plane.2. Because the function is a vertically stretched absolute value function, draw a 'V' shape starting at this vertex.3. The lines from the vertex should have a slope of 3, rising sharply at a 3:1 ratio (up 3 units for every 1 unit horizontally).
05
Check Symmetry and Shape
The graph of \( f(x) = 3|x-2|-3 \) should be symmetric with respect to the line \( x=2 \) because it is based on an absolute value function, which is symmetric around its vertex.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Shift
In the context of the function \( f(x) = 3|x-2|-3 \), the term \( x-2 \) represents a horizontal shift. Typically, the function \( f(x) = |x| \) has its vertex at the origin, \( (0,0) \). However, adjusting the term within the absolute value changes its position horizontally on the graph.
- The expression \( |x-2| \) tells us that the entire graph shifts 2 units to the right.
- This is because the subtraction of 2 within the absolute value moves the base absolute value function, \( |x| \), rightward along the x-axis.
Vertical Stretch
The coefficient of 3 in the function \( f(x) = 3|x-2|-3 \) indicates a vertical stretch. In general, a function \( f(x) = a|x| \) experiences a vertical stretch when \( a > 1 \).
- This multiplier changes how the graph looks without altering its overall direction or symmetry.
- Specifically, for \( a = 3 \), the slope of the lines forming the 'V' shape becomes steeper, rising at a rate of 3 units vertically for every 1 unit horizontally.
- The applied multiplier creates more pronounced angles, emphasizing the sharpness of the graph's appearance.
Vertex of a Function
For the function \( f(x) = 3|x-2|-3 \), the vertex is essentially the "corner" or the lowest point (for upward-opening graphs) on the graph. In this case, the vertex is found by identifying the shifts:
- Start at the origin based on the classic absolute value function, \( f(x) = |x| \), which is \( (0,0) \).
- Then, apply a horizontal shift to the right by 2 units, making the new position \( (2,0) \).
- Next, you account for the vertical shift downwards by 3 units. Thus, \( (2,0) \) becomes \( (2,-3) \).
Graph Symmetry
Graph symmetry is an essential concept, especially for absolute value functions. The function \( f(x) = 3|x-2|-3 \) demonstrates symmetry centered around its vertex due to its mathematical nature.
- The graph is symmetric along the vertical line defined by \( x = 2 \), which corresponds to the x-coordinate of the vertex.
- This means each point to the left of the line \( x=2 \) has a mirror image on the right, and vice versa.
- Symmetry in such functions helps predict the path of the graph and simplifies the plotting process.
