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Figure 1 shows the graph of $y = |2x|$ - AQA - A-Level Maths Mechanics - 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 Mechanics - 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 start by identifying the vertex of the V shape.
Setting the expression inside the absolute value to zero: egin{align*} 3x - 6 &= 0 \ 3x &= 6 \ x &= 2 \\ ext{Thus, the vertex is at } (2, 0). \end{align*}
Next, we determine the direction of the arms of the V shape. As is positive for , the line has a positive slope to the right (slope of 3), and as approaches negative infinity, the shifts the graph downwards. Therefore, the graph increases to the left in a symmetric manner. Finally, sketching this onto Figure 1, we ensure the V-shape intersects the y-axis at:
This shows that the graph intersects the y-axis at (0, 6) and is symmetric about the line x = 2.
Step 2
4 (b) Find the coordinates of the points of intersection of the two graphs.
Answer
To find the points of intersection, we set the equations equal to each other:
This leads us to consider two cases based on the definitions of absolute values:
Case 1:
- Rearranging gives:
so,
Substituting back into :
Thus, one point of intersection is (6, 12).
Case 2:
- Rearranging gives:
which simplifies to:
so,
Substituting back into :
Thus, the second point of intersection is (1.2, 2.4).
Conclusion
The coordinates of the points of intersection are (6, 12) and (1.2, 2.4).