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Figure 1 shows the graph of $y = |2x|$ - AQA - A-Level Maths Pure - Question 4 - 2021 - Paper 2
Question 4
Figure 1 shows the graph of $y = |2x|$. 4 (a) On Figure 1 add a sketch of the graph of $y = |3x - 6|$. 4 (b) Find the coordinates of the points of intersection of... show full transcript
Worked Solution & Example Answer:Figure 1 shows the graph of $y = |2x|$ - AQA - A-Level Maths Pure - Question 4 - 2021 - Paper 2
Step 1
4 (a) On Figure 1 add a sketch of the graph of $y = |3x - 6|$
Answer
To sketch the graph of , we first identify the vertex. Setting the expression inside the absolute value to zero, we solve:
x = 2$$ The vertex of the graph is at the point (2, 0). The graph will then be a V-shape, opening upwards, with the arms of the graph increasing from the vertex. Next, we determine a few points for drawing the graph: - When $x = 0$: $$y = |3(0) - 6| = | -6| = 6 \ ext{(point: (0, 6))}$$ - When $x = 4$: $$y = |3(4) - 6| = |12 - 6| = 6 \ ext{(point: (4, 6))}$$ Thus, the graph touches the y-axis at (0, 6) and (4, 6) on either side of the vertex (2, 0). We plot these points and connect them to form the V-shape, ensuring the apex is at (2, 0) and symmetrically extending up to the positive y-axis.Step 2
4 (b) Find the coordinates of the points of intersection of the two graphs
Answer
To find the points of intersection between and , we set the equations equal to each other:
We can solve this equation by considering different cases based on the expressions inside the absolute values.
Case 1: When :
Calculate for :
Thus, one point of intersection is (6, 12).
Case 2: When :
2x = -3x + 6 \\ 5x = 6 \\ x = rac{6}{5} = 1.2
Calculate for :
Thus, the second point of intersection is (1.2, 2.4).
In summary, the coordinates of the points of intersection are:
- (6, 12)
- (1.2, 2.4)