Without the graph, it is impossible to determine the exact rule that represents the function. To answer this question, I need to see the graph or be provided with information about it, such as:
- Specific points on the graph: Knowing several (x, y) coordinates that the function passes through is essential.
- Type of function: Is it a linear function, a quadratic function, an exponential function, a trigonometric function, or something else?
- Characteristics of the graph: Does it have a constant slope? Does it curve? Does it have asymptotes? Does it have any intercepts (x and y)?
Here's how you would determine the rule if you had the graph:
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Identify the Type of Function: Look at the overall shape of the graph to determine the function type.
- Linear: A straight line follows the form y = mx + b, where m is the slope and b is the y-intercept.
- Quadratic: A parabola follows the form y = ax2 + bx + c.
- Exponential: A curve that either increases or decreases rapidly, following the form y = abx.
- Other functions: Consider trigonometric, logarithmic, or polynomial functions depending on the graph's shape.
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Find Key Points/Characteristics: Identify key features of the graph.
- Linear: Find two points to calculate the slope m and identify the y-intercept b.
- Quadratic: Find the vertex, the y-intercept, and any x-intercepts to help determine the coefficients a, b, and c.
- Exponential: Find the y-intercept (initial value) and look for a point to help determine the base b.
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Write the Equation: Once you know the type of function and have identified key parameters, write the equation that represents the graph.
Example:
Suppose the graph is a straight line that passes through the points (0, 2) and (1, 4).
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Type: Linear function.
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Key Points:
- y-intercept (b) = 2 (because the line passes through (0, 2))
- Slope (m) = (4 - 2) / (1 - 0) = 2
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Equation: y = 2x + 2
Since the graph is not provided, I cannot give a specific answer.