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Increasing, decreasing and constant functions

Algebra, grade 8 Lesson 6 of 27Learning

Increasing, decreasing and constant functions

Increasing, decreasing, constant
On the x\displaystyle x-axis the numbers grow to the right.

A graph is read from left to right – the way x\displaystyle x grows – asking what happens to y\displaystyle y:

  • Increasing – as x\displaystyle x grows, y\displaystyle y grows too. The graph goes up.
  • Decreasing – as x\displaystyle x grows, y\displaystyle y gets smaller. The graph goes down.
  • Constant – as x\displaystyle x grows, y\displaystyle y does not change. The graph is level.
Example: the temperature during a day

The graph shows the temperature in a town during one day.
The x\displaystyle x-axis: the hour, from midnight to midnight. The y\displaystyle y-axis: the temperature in degrees.

  • From midnight until 5 the temperature decreases, from 12 degrees to 8 degrees.
  • From 5 until 14 it increases, up to 26 degrees.
  • From 14 until midnight it decreases again.
T(x)\displaystyle T(x)
(5,8)\displaystyle (5,8)
(14,26)\displaystyle (14,26)
Intervals of increase and decrease
An interval is a set of values of x\displaystyle x: “increasing for 5<x<14\displaystyle 5<x<14”, not “for 8<y<26\displaystyle 8<y<26”.

An interval of increase is a stretch of the x\displaystyle x-axis where the function increases, and an interval of decrease – a stretch where it decreases. They are written in words or with the signs <\displaystyle < and >\displaystyle >.

  • 5<x<14\displaystyle 5<x<14 means: x\displaystyle x is greater than 5 and less than 14 – all the numbers between 5 and 14.
  • For the temperature – the interval of increase: 5<x<14\displaystyle 5<x<14.
  • The intervals of decrease: 0<x<5\displaystyle 0<x<5 and 14<x<24\displaystyle 14<x<24.
Maximum and minimum points

A maximum point is a point where the function turns from increasing to decreasing – a peak of the graph. A minimum point is a point where it turns from decreasing to increasing – a valley.

  • For the temperature, (14,26)\displaystyle (14,26) is a maximum point: at 14 it is hottest, 26 degrees.
  • (5,8)\displaystyle (5,8) is a minimum point: at 5 it is coldest, 8 degrees.
  • A point is written as an ordered pair. Its y\displaystyle y is the greatest value (at a maximum) or the least (at a minimum).
An interval where the function is constant

This is the graph of Noa’s bike ride: x\displaystyle x – the time in hours, y\displaystyle y – the distance from home in kilometres.

d(x)\displaystyle d(x)
  • Intervals of increase – Noa moves away from home: 0<x<1\displaystyle 0<x<1 and 1.5<x<2.5\displaystyle 1.5<x<2.5.
  • An interval where the function is constant – Noa rests: 1<x<1.5\displaystyle 1<x<1.5.
  • The interval of decrease – Noa rides back home: 2.5<x<4\displaystyle 2.5<x<4.
  • (2.5,18)\displaystyle (2.5,18) is a maximum point: after two and a half hours Noa is farthest from home.
Increase and decrease in a table
A table shows only its columns. Here we assume that between one column and the next the function goes on in the same direction.

A table of values shows increase and decrease too: go through the columns in order, from left to right, and check whether y\displaystyle y grows or shrinks.

x\displaystyle x−3\displaystyle -3−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 3
y\displaystyle y5\displaystyle 50\displaystyle 0−3\displaystyle -3−4\displaystyle -4−3\displaystyle -30\displaystyle 05\displaystyle 5
  • Until x=0\displaystyle {x}={0} the values of y\displaystyle y get smaller: the function decreases.
  • After x=0\displaystyle {x}={0} they grow: the function increases.
  • (0,−4)\displaystyle (0,-4) is a minimum point.
Exercise 1

Mark the maximum and minimum points of g\displaystyle g.
Write:

  • for each point you marked – maximum or minimum;
  • the intervals of increase and the intervals of decrease of g\displaystyle g.
g(x)\displaystyle g(x)

Exercise 2

A lift left the ground floor, floor 0.
The graph shows which floor the lift was at, at every moment.
The x\displaystyle x-axis: the time in minutes. The y\displaystyle y-axis: the floor.

  • Write the intervals of increase, the intervals of decrease and the intervals where the function is constant.
  • What was the lift doing in each interval where the function is constant?
  • What is the highest floor the lift reached, and when was it there?
h(x)\displaystyle h(x)
The highest floor

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Exercise 3
    x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
    y\displaystyle y7\displaystyle 72\displaystyle 2−1\displaystyle -1−2\displaystyle -2−1\displaystyle -12\displaystyle 27\displaystyle 7
    • In which interval does the function decrease, and in which does it increase?
    • Find its minimum point.
    Exercise 4

    Sketch the graph of a function that has all these properties:

    • it increases for x<1\displaystyle x<1;
    • (1,5)\displaystyle (1,5) is a maximum point;
    • it decreases for 1<x<4\displaystyle 1<x<4;
    • it is constant for x>4\displaystyle x>4.