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Figure 2 shows a sketch of the graph with equation y = 2|x + 4| - 5 The vertex of the graph is at the point P, shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2
Question 12
Figure 2 shows a sketch of the graph with equation y = 2|x + 4| - 5 The vertex of the graph is at the point P, shown in Figure 2. (a) Find the coordinates of P. ... show full transcript
Worked Solution & Example Answer:Figure 2 shows a sketch of the graph with equation y = 2|x + 4| - 5 The vertex of the graph is at the point P, shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2
Step 1
Find the coordinates of P.
Answer
To find the coordinates of point P, we need to determine the vertex of the given equation y = 2|x + 4| - 5. The vertex occurs when the expression inside the absolute value is zero.
Setting |x + 4| = 0 gives us:
Substituting this value back into the equation to find y:
Thus, the coordinates of P are (-4, -5).
Step 2
Solve the equation 3x + 40 = 2|x + 4| - 5.
Answer
To solve the equation, first, simplify it:
3x + 45 = 2|x + 4| \\ \Rightarrow |x + 4| = \frac{3x + 45}{2}$$ We consider two cases for the absolute value: **Case 1:** When `x + 4 ≥ 0`, then `|x + 4| = x + 4`. Substituting this into the equation gives: $$x + 4 = \frac{3x + 45}{2} \\ 2(x + 4) = 3x + 45 \\ 2x + 8 = 3x + 45 \\ x = -37$$ Since `-37 < -4`, this solution is invalid for this case. **Case 2:** When `x + 4 < 0`, then `|x + 4| = -(x + 4)`. Substituting gives: $$-(x + 4) = \frac{3x + 45}{2} \\ -2(x + 4) = 3x + 45 \\ -2x - 8 = 3x + 45 \\ -5x = 53 \\ x = -\frac{53}{5} = -10.6$$ This solution is valid since `-10.6 < -4`.Step 3
Find the range of possible values of a, writing your answer in set notation.
Answer
For line l: y = ax to intersect the graph of y = 2|x + 4| - 5, we require at least one solution for the equation:
At the vertex x = -4, we have:
To ensure an intersection at this y-value,
\Rightarrow a = \frac{-5}{x}$$ Considering the behavior of `y = 2|x + 4| - 5`, we analyze the slopes: - For intersections with a line to happen, the slope `a` must be less than the slope of the line at `x = -4`: $$a < 2 \\ \Rightarrow a \in (-\infty, 2)$$ Furthermore, for the equation to yield two solutions at lesser slopes: - When `x < -4`: The slope must be greater than the steep slope of the graph as it increases from the vertex: $$a > 2$$ Combining these gives us the set notation: $$\{a: a \leq 2.5 \} \cup \{a: a > 2 \}$$