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Practice: the linear function

Algebra, grade 8 Lesson 13 of 27Practice

Practice: the linear function

1

In which tables is the rate of change constant?
Where it is constant, write it.

a.

x\displaystyle x01234
y\displaystyle y−3\displaystyle -3−1\displaystyle -1135

b.

x\displaystyle x12345
y\displaystyle y25101726

c.

x\displaystyle x0369
y\displaystyle y10864
2

Which of the points are on the line y=−3x+5\displaystyle {y}={-3x+5}?

  • (1,2)\displaystyle (1,2)
  • (−2,11)\displaystyle (-2,11)
  • (3,−3)\displaystyle (3,-3)
  • (0,5)\displaystyle (0,5)
3

Work out the slope of the line through the two points.

a
(1,2)\displaystyle (1,2), (3,8)\displaystyle (3,8)

    / – 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
    (−2,5)\displaystyle (-2,5), (2,−3)\displaystyle (2,-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

      c
      (0,−1)\displaystyle (0,-1), (4,1)\displaystyle (4,1)

        / – 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
        (−5,4)\displaystyle (-5,4), (1,4)\displaystyle (1,4)

          / – 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

          4

          Here are some lines:

          • y=4x−1\displaystyle {y}={4x-1}
          • y=−x+2\displaystyle {y}={-x+2}
          • y=3−12x\displaystyle {y}={3-\frac{1}{2}x}
          • y=4x+6\displaystyle {y}={4x+6}
          • y=2\displaystyle {y}={2}

          For each line, write its slope and where it meets the y\displaystyle y-axis.
          Which lines are parallel to each other?
          Which lines go down?

          5

          Work out where the lines y=12x−2\displaystyle {y}={\frac{1}{2}x-2} and y=−x−3\displaystyle {y}={-x-3} meet the axes.
          Mark the points.

          y=12x−2\displaystyle y=\frac{1}{2}x-2
          y=−x−3\displaystyle y=-x-3

          6

          Find where each line meets the x\displaystyle x-axis.

          a
          y=4x−10\displaystyle {y}={4x-10}
            b
            y=−23x+4\displaystyle {y}={-\frac{2}{3}x+4}
              7

              Write the formulas of the lines f\displaystyle f and g\displaystyle g, from the graph.

              f(x)= ?\displaystyle f(x)=\,?
              g(x)= ?\displaystyle g(x)=\,?

              8

              A line passes through the point P(−1,1)\displaystyle P(-1,1), and its slope is 4.
              Find its formula.

              P=(−1,1)\displaystyle P=(-1,1)

              9

              A line passes through the points A(2,−1)\displaystyle A(2,-1) and B(6,1)\displaystyle B(6,1).
              Find its formula.

              A=(2,−1)\displaystyle A=(2,-1)
              B=(6,1)\displaystyle B=(6,1)

              10

              Find the formula of the line parallel to y=−2x+1\displaystyle {y}={-2x+1} through P(−2,3)\displaystyle P(-2,3).

              y=−2x+1\displaystyle y=-2x+1
              P=(−2,3)\displaystyle P=(-2,3)

              11

              Mark the point where the lines y=x−1\displaystyle {y}={x-1} and y=−12x+5\displaystyle {y}={-\frac{1}{2}x+5} meet.
              Show by substituting that it is on both lines.

              y=x−1\displaystyle y=x-1
              y=−12x+5\displaystyle y=-\frac{1}{2}x+5

              12

              Renting a bike costs 15 ₪ to hire, and 8 ₪ more for every hour.

              • Write the formula of the price y\displaystyle y (in shekels) by the number of hours x\displaystyle x.
              • How much do 3 hours cost?
              • Dana paid 71 ₪. Write an equation and solve it: for how many hours did she rent a bike?

              3 hours

                / – 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

                13

                Two candles were lit together.
                Candle A is 24 cm tall, and gets 2 cm shorter every hour.
                Candle B is 18 cm tall, and gets 1 cm shorter every hour.
                On the graph: f\displaystyle f – the height of candle A, g\displaystyle g – the height of candle B, against the time in hours.

                • Write the formulas of f\displaystyle f and g\displaystyle g.
                • Mark the point where the graphs meet.
                • What does it mean: after how many hours are the candles the same height, and what is the height?
                • Which candle will burn out first, and after how many hours?
                f(x)= ?\displaystyle f(x)=\,?
                g(x)= ?\displaystyle g(x)=\,?

                The same height: after how many hours

                  / – 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

                  That height (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

                    Which candle burns out first
                    It burns out after (hours)

                      / – 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