Match each graph with the function it represents.
Understand the Problem
The question asks us to match the graphs with their corresponding functions. Each graph (A, B, C, D, E) needs to be associated with its correct functional representation.
Answer
* Graph A: $y=\sqrt{x}$ * Graph B: $y=\sqrt[3]{x}$ * Graph C: $y=\sqrt{x-2}$ * Graph D: $y=3^{-x}$ * Graph E: $y=4^{-x}$
Answer for screen readers
Based on the graphs and their properties, here's the matching:
- Graph A: $y=\sqrt{x}$
- Graph B: $y=\sqrt[3]{x}$
- Graph C: $y=\sqrt{x-2}$
- Graph D: $y=3^{-x}$
- Graph E: $y=4^{-x}$
Steps to Solve
- Analyze Graph A
Graph A starts at $x=0$ and increases gradually. This looks like a square root function of the form $y = \sqrt{x}$.
- Analyze Graph B
Graph B is defined for all $x$ and passes through the origin (0, 0). It increases gradually, which suggests a cube root function, $y = \sqrt[3]{x}$.
- Analyze Graph C
Graph C is similar to Graph A, starting at $x=0$ and increasing. However, it's horizontally shifted to the right. Thus, it should correspond to the form $y = \sqrt{x-c}$, where $c$ is a constant.
- Analyze Graph D
Graph D is a decreasing function and appears to level off as x increases. This suggests a negative exponential decay function.
- Analyze Graph E
Graph E is a decreasing function, similar to graph D. It also appears to be leveling off as $x$ increases, thus it is an exponential decay function as well.
Based on the graphs and their properties, here's the matching:
- Graph A: $y=\sqrt{x}$
- Graph B: $y=\sqrt[3]{x}$
- Graph C: $y=\sqrt{x-2}$
- Graph D: $y=3^{-x}$
- Graph E: $y=4^{-x}$
More Information
The base of the exponential function determines how quickly the graph decreases. A larger base (like 4 in Graph E) makes the graph decrease faster compared to a smaller base (like 3 in Graph D).
Tips
A common mistake is confusing the square root and cube root functions. Also, students might mix up exponential decay functions with different bases. Paying attention to the rate of increase or decrease and the domain of each function can help avoid these errors.
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