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Sketch the graph of y = 2x^2 - 8x - 5 showing the coordinates of the turning point and the exact coordinates of any intercepts with the coordinate axes. - Edexcel - GCSE Maths - Question 22 - 2019 - Paper 1

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Sketch-the-graph-of--y-=-2x^2---8x---5--showing-the-coordinates-of-the-turning-point-and-the-exact-coordinates-of-any-intercepts-with-the-coordinate-axes.-Edexcel-GCSE Maths-Question 22-2019-Paper 1.png

Sketch the graph of y = 2x^2 - 8x - 5 showing the coordinates of the turning point and the exact coordinates of any intercepts with the coordinate axes.

Worked Solution & Example Answer:Sketch the graph of y = 2x^2 - 8x - 5 showing the coordinates of the turning point and the exact coordinates of any intercepts with the coordinate axes. - Edexcel - GCSE Maths - Question 22 - 2019 - Paper 1

Step 1

Showing the intercepts

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Answer

To find the y-intercept, set x = 0:

y=2(0)2−8(0)−5=−5.y = 2(0)^2 - 8(0) - 5 = -5.

Thus, the y-intercept is at (0, -5).

To find the x-intercepts, set y = 0:

0=2x2−8x−5.0 = 2x^2 - 8x - 5.

Using the quadratic formula:

x=−b±b2−4ac2a=8±(−8)2−4(2)(−5)2(2)=8±64+404=8±1044=8±2264=2±262.x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4(2)(-5)}}{2(2)} = \frac{8 \pm \sqrt{64 + 40}}{4} = \frac{8 \pm \sqrt{104}}{4} = \frac{8 \pm 2\sqrt{26}}{4} = 2 \pm \frac{\sqrt{26}}{2}.

Thus, the x-intercepts are at

(2−262,0) and (2+262,0).\left( 2 - \frac{\sqrt{26}}{2}, 0 \right) \text{ and } \left( 2 + \frac{\sqrt{26}}{2}, 0 \right).

Step 2

Finding the turning point

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Answer

To find the turning point, first calculate the vertex of the parabola:

The x-coordinate is given by

x=−b2a=−−82(2)=2.x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2.

Substituting x = 2 into the equation gives:

y=2(2)2−8(2)−5=8−16−5=−13.y = 2(2)^2 - 8(2) - 5 = 8 - 16 - 5 = -13.

Therefore, the turning point is at (2, -13).

Step 3

Sketching the graph

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Answer

Sketch the graph of the function, marking the intercepts and the turning point:

  • The y-intercept at (0, -5).
  • The x-intercepts at approximately (0.9, 0) and (3.1, 0).
  • The turning point at (2, -13).

Ensure that the parabola opens upwards and that the turning point is shown clearly on the graph.

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