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The vertex: the greatest height and the least value

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

The vertex: the greatest height and the least value

Midway between two times with the same height

The vertex is on the axis of symmetry, and so midway between any two values of x\displaystyle x with the same y\displaystyle y.

A ball is thrown from the ground, and its height in metres after t\displaystyle t seconds is h=−5t2+30t\displaystyle {h}={-5t^{2}+30t}.

t\displaystyle t0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 45\displaystyle 56\displaystyle 6
h\displaystyle h0\displaystyle 025\displaystyle 2540\displaystyle 4045\displaystyle 4540\displaystyle 4025\displaystyle 250\displaystyle 0
  • The ball passes the height of 25 metres twice: at t=1\displaystyle {t}={1} and at t=5\displaystyle {t}={5}.
  • Midway between them, at t=1+52=3\displaystyle {t}={\frac{1+5}{2}}={3}, it is at its greatest height: 45 metres.
  • The zeros, t=0\displaystyle {t}={0} and t=6\displaystyle {t}={6}, give the same middle too.
x=x1+x22\displaystyle x=\frac{x_1+x_2}{2}
The formula of the axis of symmetry

By the quadratic formula, the zeros are −b+b2−4ac2a\displaystyle \frac{-b+\sqrt{b^{2}-4ac}}{2a} and −b−b2−4ac2a\displaystyle \frac{-b-\sqrt{b^{2}-4ac}}{2a}. Added together, the roots cancel, and their middle is −b2a\displaystyle \frac{-b}{2a}. That is the vertex’s x\displaystyle x:

x=−b2a\displaystyle x=-\frac{b}{2a}

The vertex’s y\displaystyle y comes from substituting this x\displaystyle x in the formula.

For the ball: a=−5\displaystyle {a}={-5}, b=30\displaystyle {b}={30}, so t=−302⋅(−5)=3\displaystyle {t}={-\frac{30}{2\cdot(-5)}}={3}, and h=−5⋅32+30⋅3=45\displaystyle {h}={-5\cdot3^{2}+30\cdot3}={45}.

The greatest height of a jet

In a fountain in the square, the water comes out of a pipe one metre high.
The jet’s height, in metres, x\displaystyle x metres from the pipe: y=−0.5x2+3x+1\displaystyle {y}={-0.5x^{2}+3x+1}

  • a=−0.5\displaystyle {a}={-0.5}, b=3\displaystyle {b}={3}, so x=−32⋅(−0.5)=31=3\displaystyle {x}={-\frac{3}{2\cdot(-0.5)}}={\frac{3}{1}}={3}.
  • y=−0.5⋅32+3⋅3+1=−4.5+9+1=5.5\displaystyle {y}={-0.5\cdot3^{2}+3\cdot3+1}={-4.5+9+1}={5.5}.

The jet reaches its greatest height, 5.5 metres, 3 metres from the pipe.

y=−0.5x2+3x+1\displaystyle y=-0.5x^{2}+3x+1
V=(3,5.5)\displaystyle V=(3,5.5)
A least value: saving fuel

The same formula gives a lowest point too, when a>0\displaystyle a>0.

A car’s fuel use, in litres per 100 km, when it drives at v\displaystyle v km/h:
c=0.002v2−0.24v+12\displaystyle {c}={0.002v^{2}-0.24v+12}

  • v=−−0.242⋅0.002=0.240.004=60\displaystyle {v}={-\frac{-0.24}{2\cdot0.002}}={\frac{0.24}{0.004}}={60} km/h.
  • c=0.002⋅602−0.24⋅60+12=7.2−14.4+12=4.8\displaystyle {c}={0.002\cdot60^{2}-0.24\cdot60+12}={7.2-14.4+12}={4.8} litres.

At 60 km/h the car uses the least fuel: 4.8 litres per 100 km. Faster – the air resists more; slower – the engine runs in a low gear, which wastes fuel.

Exercise 1

A ball is kicked straight up from the ground.
Its height, in metres, t\displaystyle t seconds after the kick: h=−5t2+40t\displaystyle {h}={-5t^{2}+40t}

a

After how many seconds is the ball at its 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

    b

    What is 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

      Exercise 2

      The jet of a garden sprinkler.
      The water’s height, in metres, x\displaystyle x metres from the sprinkler: y=−0.25x2+2x+0.5\displaystyle {y}={-0.25x^{2}+2x+0.5}

      a

      How far from the sprinkler is the jet at its 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

        b

        What is the jet’s 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

          Exercise 3

          A skateboard ramp shaped as a half-pipe, 12 metres wide.
          The ramp’s height, in metres, x\displaystyle x metres from its edge: y=0.1x2−1.2x+3.6\displaystyle {y}={0.1x^{2}-1.2x+3.6}

          a

          How far from the edge is the ramp’s lowest point?

            / – 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

            b

            What is the ramp’s height at that point?

              / – 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 4

              A sensor measured the height of a ball thrown upwards.
              After 0.5 seconds and after 2.5 seconds the ball was at the same height, 11.25 metres.

              a

              After how many seconds was the ball at its 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

                b

                The ball’s height, in metres, t\displaystyle t seconds after the throw: h=−5t2+15t+5\displaystyle {h}={-5t^{2}+15t+5}
                What is 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