Linear functions form the foundation of calculus, with their constant rate of change. , the key feature, measures steepness and direction. Understanding linear functions is crucial for grasping more complex relationships in mathematics.
Polynomial functions expand on linear concepts, introducing higher degrees and varied behaviors. These functions exhibit distinct characteristics based on their and coefficients. Mastering polynomials prepares you for analyzing more intricate mathematical relationships and real-world applications.
Linear Functions
Slope of linear functions
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Measures the rate of change or steepness of a
Calculated as the change in y-values divided by the change in x-values ΔxΔy
Remains constant for a
Positive indicates an increasing function (line rises from left to right)
Negative slope indicates a decreasing function (line falls from left to right)
Zero slope represents a horizontal line (no change in y-values)
Steeper lines have larger absolute values of slope
Polynomial Functions
Polynomial function characteristics
Consist of terms with non-negative integer exponents
is the highest exponent of the variable
Linear functions are first-degree polynomials (ax+b)
Quadratic functions are second-degree polynomials (ax2+bx+c)
Odd-degree polynomials:
Have at least one real ()
Exhibit opposite (x→−∞, f(x)→±∞ and x→∞, f(x)→∓∞)
Even-degree polynomials:
May not have any real roots
Exhibit the same (x→±∞, f(x)→∞ for positive or f(x)→−∞ for negative leading coefficient)
Quadratic equations and roots
In the form ax2+bx+c=0, where a=0
Roots are x-values where the function equals zero
Roots found using the : x=2a−b±b2−4ac
Roots are x-coordinates of points where the graph crosses the x-axis
(b2−4ac) determines the nature of the roots:
Positive discriminant indicates two distinct real roots
Zero discriminant indicates one repeated real root
Negative discriminant indicates two complex conjugate roots
Other Functions
Types of algebraic functions
Rational functions are ratios of two polynomials
May have vertical asymptotes where the denominator equals zero
May have horizontal asymptotes as x→±∞
Power functions are of the form f(x)=xa, where a is constant
Positive a results in an increasing function
Negative a results in a decreasing function
Even a results in a function symmetric about the y-axis
Odd a results in a function symmetric about the origin
Root functions are of the form f(x)=nx, where n is a positive integer