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Reading the graph of a quadratic process

Science and society, grade 12 (3 units) Lesson 2 of 14Learning

Reading the graph of a quadratic process

A stone thrown from a cliff

A hiker stands on a cliff 25 metres above the sea, and throws a stone upwards at a slant.
The stone’s height above the water, in metres, t\displaystyle t seconds after the throw:
h=−5t2+20t+25\displaystyle {h}={-5t^{2}+20t+25}

y=−5x2+20x+25\displaystyle y=-5x^{2}+20x+25
S=(0,25)\displaystyle S=(0,25)
V=(2,45)\displaystyle V=(2,45)
L=(5,0)\displaystyle L=(5,0)

Three points on the graph tell the stone’s story:

  • S=(0,25)\displaystyle {S}={(0,25)} – the graph crosses the y\displaystyle y-axis at the starting height: the stone sets off 25 metres up.
  • V=(2,45)\displaystyle {V}={(2,45)} – the vertex: after 2 seconds the stone is at its greatest height, 45 metres.
  • L=(5,0)\displaystyle {L}={(5,0)} – the graph crosses the x\displaystyle x-axis: after 5 seconds the height is 0 – the stone falls into the sea.
The words of the graph

Each special point of the graph has a name, and each name a meaning in the process:

On the graphFor the stone
The point where it crosses the y\displaystyle y-axisThe height at the start, t=0\displaystyle {t}={0}
The extremum – here a maximumThe greatest height, and when
A zero – where it crosses the x\displaystyle x-axisWhen the height is 0: the stone is in the water
Increasing: 0<t<2\displaystyle 0<t<2The stone rises
Decreasing: 2<t<5\displaystyle 2<t<5The stone falls
The part of the graph that has a meaning

The formula describes the stone only from the throw until it hits the water: 0≤t≤5\displaystyle 0\le t\le5.

  • The graph goes on to the left of t=0\displaystyle {t}={0} as well, and meets the x\displaystyle x-axis at t=−1\displaystyle {t}={-1} too. But a negative time is before the throw – it has no meaning.
  • After t=5\displaystyle {t}={5} the graph has no meaning either: the stone is in the water already, and does not go on falling “below zero” at that rate.

So an answer states only what really happened: “the stone hits the water after 5 seconds”.

A process with a least value

On a winter night in the desert, the temperature in degrees, t\displaystyle t hours after midnight:
T=0.5t2−4t+10\displaystyle {T}={0.5t^{2}-4t+10}

y=0.5x2−4x+10\displaystyle y=0.5x^{2}-4x+10

Here a=0.5>0\displaystyle {a}={0.5>0}, so the parabola opens upwards, and has a lowest point:

  • at midnight T=10\displaystyle {T}={10} degrees;
  • the temperature falls until 4 in the morning, when it is the lowest – 2 degrees: the vertex (4,2)\displaystyle (4,2) is a minimum;
  • then the temperature rises: at 8 in the morning it is 10 degrees again.
Exercise 1

A goalkeeper kicks a ball.
The ball’s height, in metres, x\displaystyle x seconds after the kick: y=−5x2+25x\displaystyle {y}={-5x^{2}+25x}

a

Mark the graph’s highest point and its zeros on the board, and check.

y=−5x2+25x\displaystyle y=-5x^{2}+25x

b
  • What is the ball’s greatest height, and after how many seconds?
  • How long is the ball in the air?
  • Where does the height increase, and where does it decrease?
The greatest height

    / – 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

    After how many seconds

      / – 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

      Time in the air

        / – 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 2

        A power cable hangs between two poles 12 metres apart.
        The cable’s height above the ground, in metres, x\displaystyle x metres from the first pole: y=0.25x2−3x+28\displaystyle {y}={0.25x^{2}-3x+28}
        Mark the cable’s lowest point on the board, and check.
        How tall are the poles? What is the cable’s least height, and how far from the first pole? A lorry 4 metres high drives under the cable – is it safe?

        y=0.25x2−3x+28\displaystyle y=0.25x^{2}-3x+28

        The poles’ height

          / – 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

          The least height

            / – 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

            The distance from the first pole

              / – 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

              Pupils at a science club launch a water rocket from the ground.
              The rocket’s height, in metres, x\displaystyle x seconds after the launch: y=−5x2+40x\displaystyle {y}={-5x^{2}+40x}
              Read from the graph:

              • What is the rocket’s greatest height, and when does it reach it?
              • When does the rocket land?
              • Where does the height increase, and where does it decrease?
              • Which part of the graph has a meaning?
              y=−5x2+40x\displaystyle y=-5x^{2}+40x
              The greatest height

                / – 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

                When it gets there

                  / – 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

                  Landing

                    / – 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