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Properties of Functions

Maths: Edexcel A Level Pure Maths, topic 2.6.1

Properties of Functions

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

Vertices

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.
Axis of symmetry

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.
Axis of symmetry

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)2−b24a2+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}
Average rate of change

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)x2−x1\frac{f(x_2)-f(x_1)}{x_2 - x_1}
Increasing or decreasing

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.

Test yourself on Properties of Functions

  • What is the equation of the axis of symmetry for a parabola of the equation 5x2−3x+175x^2 - 3x+17?

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