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Sketch the region defined by the inequalities y ≤ (1 − 2x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths: Pure - Question 4 - 2019 - Paper 3
Question 4
Sketch the region defined by the inequalities y ≤ (1 − 2x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R.
Worked Solution & Example Answer:Sketch the region defined by the inequalities y ≤ (1 − 2x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths: Pure - Question 4 - 2019 - Paper 3
Step 1
y ≤ (1 − 2x)(x + 3)
Answer
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Begin by determining the equation of the quadratic:
This expands to:
- Identify the vertex and roots to sketch the curve:
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The vertex occurs at the point where the derivative is zero.
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Solve for x by using the vertex formula or completing the square.
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For our parabola, the x-coordinate of the vertex can be found using
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Calculate y at this x-value.
- Determine the x-intercepts by solving:
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Use the quadratic formula to find the roots.
- Plot the curve with the correct orientation above the x-axis and label the vertex and intercepts.
Step 2
y − x ≤ 3
Answer
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Rearranging gives:
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This is a straight line with a slope of 1 and a y-intercept of 3.
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Identify points on the line by substituting values for x:
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For x = 0, y = 3 (point (0, 3))
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For x = -3, y = 0 (point (-3, 0))
- Sketch the line and shade the region below it since we need y less than or equal to the line.
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Step 3
Shade the region R
Answer
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Identify the area of intersection between the parabola and the straight line.
- Shade the regions that satisfy both inequalities:
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Keep the area below the line y = x + 3.
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Ensure the area remains below the quadratic curve as dictated by y ≤ (1 − 2x)(x + 3).
- Label the shaded region as R for clarity.