To design a set of functions that includes 10 functions with the specified combinations of linear, quadratic, and exponential functions, we can build upon your existing functions while ensuring we have the correct types. Here’s a list of functions divided into the three required categories:
Quadratic Functions
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Original Function 1: \( x^2 + y^2 = 25 \)
- This represents a circle centered at the origin with a radius of 5.
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Original Function 2: \( x^2 + (y - 2)^2 = 25 {y \le -2.1} \)
- This represents part of a circle (the lower half) centered at \((0, 2)\) with a radius of 5.
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Original Function 3: \((x - 2)^2 + (y - 2)^2 \le 0.5\)
- This represents a circle centered at \((2, 2)\) with a very small radius of approximately \( \sqrt{0.5} \).
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New Function 1: \( y = -x^2 + 4 \)
- This represents a downward-opening parabola with vertex at (0, 4).
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New Function 2: \( y = 3x^2 - 6 \)
- This represents an upward-opening parabola with vertex at (0, -6).
Linear Functions
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Original Function 4: \( y = 12 { -4 \le x \le 4 } \)
- This is a horizontal line segment at \(y = 12\) from \(x = -4\) to \(x = 4\).
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Original Function 5: \( x = -4 { 5 \le y \le 12 } \)
- This is a vertical line segment at \(x = -4\) from \(y = 5\) to \(y = 12\).
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Original Function 6: \( x = 4 { 5 \le y \le 12 } \)
- This is a vertical line segment at \(x = 4\) from \(y = 5\) to \(y = 12\).
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New Function 1: \( y = 2x + 1 \)
- This represents a linear function with a slope of 2 and a y-intercept of 1.
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New Function 2: \( y = -\frac{1}{2}x + 5 \)
- This represents a linear function with a slope of -0.5 and a y-intercept at 5.
Exponential Functions
- New Function 1: \( f(x) = 2^x \)
- This represents an exponential growth function.
- New Function 2: \( f(x) = e^{-x} \)
- This represents an exponential decay function.
This collection of 12 functions includes 6 quadratic functions, 6 linear functions, and 2 exponential functions, adhering to your requirements and categories. Adjust or remove as necessary to fit specific conditions or requirements.