This problem presents a scenario where the side length of a larger square picture frame is dependent on the area of a smaller square picture frame. We are given a function, S(x) = 3√ x-1, that models this relationship. Our task is to identify the graph that accurately represents this function....
Key Concepts
To solve this problem, we need to understand the following concepts:
1. Square Area
The area of a square is calculated by squaring the side length. If 's' is the side length of a square, its area 'A' is given by: A = s²
2. Cube Root
The cube root of a number is the value that, when multiplied by itself three times, results in the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Mathematically, we represent the cube root using the radical symbol (√) with a small 3 above it: ³√.
3. Function Graphs
A graph is a visual representation of a function. It shows how the output (y-value) of a function changes as the input (x-value) changes. The graph of a function is typically plotted on a coordinate plane, with the x-axis representing the input values and the y-axis representing the output values.
Analyzing the Function
Let's break down the given function, S(x) = 3√ x-1:
1. Input Variable (x)
The input variable 'x' represents the area of the smaller square frame in square inches.
2. Cube Root (√ x-1)
The cube root of (x-1) indicates that we are taking the cube root of the area of the smaller frame minus 1. This suggests a slight shift in the relationship between the area and the side length of the larger frame.
3. Constant Multiplier (3)
The constant multiplier '3' implies that the side length of the larger frame is three times the cube root of (x-1). This indicates a scaling factor in the relationship.
Identifying the Correct Graph
To identify the correct graph, we can consider the following steps:
1. Input Values
Choose a few values for 'x', which represents the area of the smaller frame. For example, we can choose x = 1, x = 2, x = 5, and x = 10.
2. Calculate Output Values
Calculate the corresponding output values for 'S(x)' using the function S(x) = 3√ x-1 for each chosen input value. For example, when x = 1, S(x) = 3√ 1-1 = 0. When x = 2, S(x) = 3√ 2-1 = 3, and so on.
3. Plot Points
Plot the calculated (x, S(x)) pairs on a coordinate plane. The x-values will represent the area of the smaller frame, and the y-values will represent the side length of the larger frame.
4. Connect the Points
Connect the plotted points with a smooth curve. This curve will represent the graph of the function S(x) = 3√ x-1.
Important Considerations
When analyzing the graph, pay attention to the following aspects:
1. Shape
The graph of S(x) = 3√ x-1 will have a specific shape, determined by the cube root function. The graph should exhibit an increasing trend, but it will not be a straight line. The curve will be smoother than a linear graph.
2. Intercept
The graph should intersect the y-axis at a specific point. This point represents the side length of the larger frame when the area of the smaller frame is 0.
3. Asymptotes
The graph may exhibit asymptotes, which are lines that the graph approaches but never touches. Asymptotes can provide insights into the behavior of the function as the input values approach infinity or negative infinity.
Conclusion
By analyzing the function S(x) = 3√ x-1, considering the key concepts, and plotting the points, we can identify the correct graph that represents the relationship between the side length of the larger frame and the area of the smaller frame. The graph will be a smooth curve that exhibits an increasing trend, with a specific intercept and potential asymptotes.