Maths 1 week 7 Summary

📚

Types of Functions

1. Linear Functions

A linear function is a function that creates a straight line when graphed. It has the form:

$$ f(x) = mx + b $$

where m is the slope and b is the y-intercept. Linear functions have a constant rate of change.

2. Quadratic Functions

A quadratic function forms a parabola when graphed. It has the form:

$$ f(x) = ax^2 + bx + c $$

where a, b, and c are constants. The vertex of the parabola represents the maximum or minimum value of the function.

3. Polynomial Functions

Polynomial functions are sums of terms consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. A general polynomial function of degree n is:

$$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $$

where a_n, a_{n-1}, ..., a_1, a_0 are constants.

4. Exponential Functions

Exponential functions have the form:

$$ f(x) = a \cdot b^x $$

where a is a constant and b is the base of the exponential. These functions grow or decay at a rate proportional to their current value.

5. Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They have the form:

$$ f(x) = \log_b(x) $$

where b is the base of the logarithm. These functions grow slowly compared to polynomial and exponential functions.

Sensitivity Comparison

The sensitivity of a function refers to how much the output value changes in response to changes in the input value. Here's a brief comparison:

• Linear functions: have a constant sensitivity.
• Quadratic functions: have a sensitivity that increases or decreases linearly.
• Polynomial functions: of higher degrees have more complex sensitivities, often increasing rapidly.
• Exponential functions: have a sensitivity that increases exponentially.
• Logarithmic functions: have a sensitivity that decreases as the input increases.

Additional Resources

For more detailed information, you can refer to the following PDF: