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Use the point (0, -12) from the table and substitute x=0 and y=-12 into the relation y = px² − 5x + q.

Mathematics

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2 parts
c(i)
-14.5
c(ii)

3.1 or −0.63.1 \text{ or } -0.6

Step-by-step solution

7 steps

Part a(i)· 2 steps

Use the table to find the values of p and q.

  1. Use the point (0,−12)(0, -12) from the table and substitute x=0x=0 and y=−12y=-12 into the relation y=px2−5x+qy = px^2 - 5x + q.

    −12=p(0)2−5(0)+q-12 = p(0)^2 - 5(0) + q −12=0−0+q-12 = 0 - 0 + q q=−12q = -12
  2. Use another point from the table, for example (4,0)(4, 0), and substitute x=4x=4, y=0y=0, and q=−12q=-12 into the relation y=px2−5x+qy = px^2 - 5x + q.

    0=p(4)2−5(4)+(−12)0 = p(4)^2 - 5(4) + (-12) 0=16p−20−120 = 16p - 20 - 12 0=16p−320 = 16p - 32 16p=3216p = 32 p=3216p = \frac{32}{16} p=2p = 2

    The values are p=2p=2 and q=−12q=-12.
    The relation is y=2x2−5x−12y = 2x^2 - 5x - 12.

Part a(ii)

Copy and complete the table.

Using the relation y=2x2−5x−12y = 2x^2 - 5x - 12, we calculate the missing y-values:

For x=−1x=-1:

y=2(−1)2−5(−1)−12y = 2(-1)^2 - 5(-1) - 12 y=2(1)+5−12y = 2(1) + 5 - 12 y=2+5−12y = 2 + 5 - 12 y=7−12=−5y = 7 - 12 = -5

For x=1x=1:

y=2(1)2−5(1)−12y = 2(1)^2 - 5(1) - 12 y=2(1)−5−12y = 2(1) - 5 - 12 y=2−5−12y = 2 - 5 - 12 y=−3−12=−15y = -3 - 12 = -15

For x=2x=2:

y=2(2)2−5(2)−12y = 2(2)^2 - 5(2) - 12 y=2(4)−10−12y = 2(4) - 10 - 12 y=8−10−12y = 8 - 10 - 12 y=−2−12=−14y = -2 - 12 = -14

For x=3x=3:

y=2(3)2−5(3)−12y = 2(3)^2 - 5(3) - 12 y=2(9)−15−12y = 2(9) - 15 - 12 y=18−15−12y = 18 - 15 - 12 y=3−12=−9y = 3 - 12 = -9

The completed table is:

x-3-2-1012345
y216-5-12-15-14-9013

Part b· 5 steps

Using scales of 2cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the relation for −3≤x≤5-3 \le x \le 5.

  1. Draw the x-axis and y-axis on graph paper.

  2. Mark the x-axis from -3 to 5, with each 2 cm representing 1 unit.

  3. Mark the y-axis to cover the range of y-values (from -15 to 21), with each 2 cm representing 5 units.

  4. Plot the points from the completed table

    (−3,21)(-3, 21), (−2,6)(-2, 6), (−1,−5)(-1, -5), (0,−12)(0, -12), (1,−15)(1, -15), (2,−14)(2, -14), (3,−9)(3, -9), (4,0)(4, 0), (5,13)(5, 13).

  5. Draw a smooth curve connecting all the plotted points.

    This curve represents the graph of y=2x2−5x−12y = 2x^2 - 5x - 12.

Part c(i)-14.5

y when x = 1.8

Use the graph to find

To find yy when x=1.8x=1.8 from the graph, locate x=1.8x=1.8 on the x-axis. Move vertically from x=1.8x=1.8 to intersect the curve. From the intersection point, move horizontally to the y-axis to read the corresponding y-value.
Using the equation y=2x2−5x−12y = 2x^2 - 5x - 12:

y=2(1.8)2−5(1.8)−12y = 2(1.8)^2 - 5(1.8) - 12 y=2(3.24)−9−12y = 2(3.24) - 9 - 12 y=6.48−9−12y = 6.48 - 9 - 12 y=6.48−21y = 6.48 - 21 y=−14.52y = -14.52

From the graph, yy would be approximately −14.5\boxed{-14.5}.

Part c(ii)

3.1 or −0.63.1 \text{ or } -0.6

x when y = -8

To find xx when y=−8y=-8 from the graph, locate y=−8y=-8 on the y-axis. Move horizontally from y=−8y=-8 to intersect the curve. From the intersection points, move vertically to the x-axis to read the corresponding x-values.
Using the equation y=2x2−5x−12y = 2x^2 - 5x - 12:

−8=2x2−5x−12-8 = 2x^2 - 5x - 12 0=2x2−5x−40 = 2x^2 - 5x - 4

Using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}:

x=−(−5)±(−5)2−4(2)(−4)2(2)x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-4)}}{2(2)} x=5±25+324x = \frac{5 \pm \sqrt{25 + 32}}{4} x=5±574x = \frac{5 \pm \sqrt{57}}{4} x1=5+574≈5+7.54984≈12.54984≈3.137x_1 = \frac{5 + \sqrt{57}}{4} \approx \frac{5 + 7.5498}{4} \approx \frac{12.5498}{4} \approx 3.137 x2=5−574≈5−7.54984≈−2.54984≈−0.637x_2 = \frac{5 - \sqrt{57}}{4} \approx \frac{5 - 7.5498}{4} \approx \frac{-2.5498}{4} \approx -0.637

From the graph, xx would be approximately 3.1 or −0.6\boxed{3.1 \text{ or } -0.6}.

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