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Sketch the region defined by the inequalities y ≤ (1 − 2)(x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths Mechanics - Question 4 - 2019 - Paper 3

Question 4

Sketch the region defined by the inequalities y ≤ (1 − 2)(x)(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 − 2)(x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths Mechanics - Question 4 - 2019 - Paper 3

Step 1

y ≤ (1 − 2)(x)(x + 3)

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Answer

To start, we need to rewrite the inequality. The expression can be simplified as:

$y ≤ -2(x^2 + 3x)$

This is a downward-opening quadratic curve with its vertex above the x-axis. We will find the vertex by using the vertex formula, where the x-coordinate of the vertex can be found using the formula: $x_v = -\frac{b}{2a}$ with ( a = -2 ) and ( b = 0 )

Thus, we have: $x_v = 0$ .

Substituting ( x = 0 ) back into the original equation gives: $y = -2(0)(0 + 3) = 0$ . So, the vertex is at (0, 0).

Next, to find the x-intercepts, we solve the equation: $y = -2x(x + 3)$ at ( y = 0 ) which gives:

( x = 0 ) and
( x + 3 = 0 ) leading us to ( x = -3 ). Thus, the intercepts on the x-axis are (0, 0) and (-3, 0).

Step 2

y − x ≤ 3

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Answer

For the second inequality, we can rewrite it as: $y ≤ x + 3$ .

This represents a straight line with a y-intercept at (0, 3) and a slope of 1. The line will pass through the points (0, 3) and (3, 0).

To find the region defined by the inequalities, we will now graph both curves: the quadratic curve and the linear function on the same axes.

Step 3

Shading and Labeling the Region R

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Answer

Once both the quadratic and linear inequalities are plotted, the region where both conditions are satisfied is the area underneath the quadratic curve and below the line (to the left of the line segment connecting (0, 3) and (3, 0)).

The area of intersection should be shaded and labeled as ( R ). Ensure all lines are solid to signify that the boundary conditions are included.

Sketch the region defined by the inequalities y ≤ (1 − 2)(x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths Mechanics - Question 4 - 2019 - Paper 3

Worked Solution & Example Answer:Sketch the region defined by the inequalities y ≤ (1 − 2)(x)(x + 3) and y − x ≤ 3 Clearly indicate your region by shading it in and labelling it R. - AQA - A-Level Maths Mechanics - Question 4 - 2019 - Paper 3

y ≤ (1 − 2)(x)(x + 3)

y − x ≤ 3

Shading and Labeling the Region R

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