2.6.1

Properties of Functions

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Properties of Functions

By defining certain properties of functions, we can get a sense of how they behave for different values of their input.

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Vertices

  • The vertex of a parabola is its turning point.
  • This vertex can either be a maximum or a minimum.
    • A “happy” parabola (of form ax2ax^2) has a vertex that is a minimum.
    • A “sad” parabola (of form ax2-ax^2) has a vertex that is a maximum.
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Axis of symmetry

  • The axis of symmetry is the line that cuts the parabola into two halves which are mirror images of each other.
  • If the parabola is vertical, the equation of the axis of symmetry has the form x=cx = c, where cc is the xx-coordinate of the vertex.
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Axis of symmetry

  • We can find the value of cc by looking at the general form of a quadratic equation y=ax2+bx+cy = ax^2 + bx + c.
  • Completing the square on this expression gives:
    • y=a[(x+b2a)2b24a2+ca]y = a\left[(x + \frac{b}{2a})^2 -\frac{b^2}{4a^2} + \frac{c}{a}\right]
  • The xx-coordinate of the vertex is b2a-\frac{b}{2a}.
  • So the axis of symmetry is given by the equation x=b2ax = -\frac{b}{2a}
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Average rate of change

  • A rate of change describes how an output quantity changes relative to the change in the input quantity.
  • The units on a rate of change are “output units per input units.”
  • To find the average rate of change of a function, we divide the change in the output value by the change in the input value.
    • Average rate of change = f(x2)f(x1)x2x1\frac{f(x_2)-f(x_1)}{x_2 - x_1}
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Increasing or decreasing

  • A function is increasing on an interval if the function values increase as the input values increase within that interval.
  • A function is decreasing on an interval if the function values decrease as the input values decrease within that interval.
  • The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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