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Practice: functions

Algebra, grade 8 Lesson 7 of 27Practice

Practice: functions

1

Is the matching a function? Explain.

  1. To each book in the library – the number of its pages.
  2. To each student – the teachers who teach them.
  3. To each number – the number twice as large.
  4. To each positive number – the numbers whose square equals it.
  5. The table:
x\displaystyle x1\displaystyle 12\displaystyle 23\displaystyle 31\displaystyle 1
y\displaystyle y4\displaystyle 45\displaystyle 56\displaystyle 67\displaystyle 7
2

The function’s formula: y=3x−2\displaystyle {y}={3x-2}.
Make a table of values for x=−2,−1,0,1,2\displaystyle {x}={-2, -1, 0, 1, 2}.

3

Find y\displaystyle y by substituting.

a

y=−2x+7\displaystyle {y}={-2x+7}, x=3\displaystyle {x}={3}

    / – 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

    y=−2x+7\displaystyle {y}={-2x+7}, x=−4\displaystyle {x}={-4}

      / – 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

      c

      y=x2−4x\displaystyle {y}={x^{2}-4x}, x=−1\displaystyle {x}={-1}

        / – 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

        d

        y=x2+3\displaystyle {y}={\frac{x}{2}+3}, x=5\displaystyle {x}={5}

          / – 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

          4

          The function’s formula: y=4x−3\displaystyle {y}={4x-3}.
          Find the x\displaystyle x at which the function takes each of the values.
          Write an equation and solve it.

          a

          y=17\displaystyle {y}={17}

          b

          y=−11\displaystyle {y}={-11}

          5

          The points A\displaystyle A, B\displaystyle B, C\displaystyle C and D\displaystyle D fit the table:

          x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 3
          y\displaystyle y−1\displaystyle -11\displaystyle 13\displaystyle 35\displaystyle 5

          Write the function’s formula.

          A=(0,−1)\displaystyle A=(0,-1)
          B=(1,1)\displaystyle B=(1,1)
          C=(2,3)\displaystyle C=(2,3)
          D=(3,5)\displaystyle D=(3,5)

          6

          On the graph of f\displaystyle f, mark the points where it meets the axes, and its maximum and minimum points.
          Write:

          • y\displaystyle y when x=2\displaystyle {x}={2};
          • for each point you marked – what it is;
          • the intervals of increase and decrease of f\displaystyle f.
          f(x)\displaystyle f(x)

          x=2\displaystyle {x}={2}

            / – 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

            7

            Is there a function whose graph holds all the points in the list? Explain with a vertical line.

            1. (1,2)\displaystyle (1,2), (2,2)\displaystyle (2,2), (3,2)\displaystyle (3,2)
            2. (2,1)\displaystyle (2,1), (2,5)\displaystyle (2,5), (3,4)\displaystyle (3,4)
            3. (−1,4)\displaystyle (-1,4), (0,1)\displaystyle (0,1), (1,4)\displaystyle (1,4)
            8

            Omer filled a bath, had a bath, and then pulled out the plug.
            The graph shows the height of the water in the bath.
            The x\displaystyle x-axis: the time in minutes. The y\displaystyle y-axis: the height of the water in cm.

            • Write the interval of increase, the interval of decrease and the interval where the function is constant. What happened in the bath in each of them?
            • How high was the water after 4 minutes?
            • When was the water 20 cm high? There are two answers.
            h(x)\displaystyle h(x)
            After 4 minutes (cm)

              / – 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

              9
              x\displaystyle x−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 45\displaystyle 5
              y\displaystyle y−5\displaystyle -50\displaystyle 03\displaystyle 34\displaystyle 43\displaystyle 30\displaystyle 0−5\displaystyle -5
              • In which interval does the function increase, and in which does it decrease?
              • Find its maximum point.
              10

              A gym charges 100 shekels to join.
              Each month costs 80 shekels more.

              • Write a formula: y\displaystyle y – how much you have paid in all after x\displaystyle x months.
              • Roni paid 580 shekels in all. For how many months did he train? Write an equation and solve it.

              11

              Match each story with its graph – p\displaystyle p, q\displaystyle q or r\displaystyle r – and explain.

              • The ball: a ball is thrown up and falls. y\displaystyle y – its height.
              • The car: a car drives at a steady speed. y\displaystyle y – the distance it has covered.
              • The phone: a phone is charging, until the battery is full. y\displaystyle y – the battery’s charge in percent.

              In all the stories x\displaystyle x is the time.

              p(x)\displaystyle p(x)
              q(x)\displaystyle q(x)
              r(x)\displaystyle r(x)
              12

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

              • it decreases for x<−2\displaystyle x<-2 and for x>2\displaystyle x>2;
              • it increases for −2<x<2\displaystyle -2<x<2;
              • (−2,−3)\displaystyle (-2,-3) is a minimum point, and (2,4)\displaystyle (2,4) is a maximum point.