Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$. The graph consists of two line segments that meet at the point $(1, a)$, where $a < 0$. One line meets the x-axis at $(3, 0)$. The other line meets the x-axis at $(-1, 0)$ and the y-axis at $(0, b)$, where $b < 0$. In separate diagrams, sketch the graph with equation a) $y = f(x)$, b) $y = f(|x|)$. Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that $f(x) = |x - 1| - 2$, find (1) the value of $a$ and the value of $b$, (2) the value of $x$ for which $f(x) = 5$.

Photo AI

Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 2 - 2005 - Paper 5

Question 2

$Figure-1-shows-part-of-the-graph-of-$y-=-f(x),-x-\in-\mathbb{R}$-Edexcel-A-Level Maths Pure-Question 2-2005-Paper 5.png$

Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$. The graph consists of two line segments that meet at the point $(1, a)$, where $a < 0$. One line me... show full transcript

Worked Solution & Example Answer:Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 2 - 2005 - Paper 5

Step 1

a) $y = f(x)$

96%

114 rated

Answer

To sketch the graph of $y = f(x)$ , we consider the piecewise nature of the function defined by the absolute value.

The function $f(x) = |x - 1| - 2$ consists of two segments:

For $x < 1$ , $f(x) = -(x - 1) - 2 = -x - 1$ .
For $x \geq 1$ , $f(x) = (x - 1) - 2 = x - 3$ .

The graph will meet the y-axis at $b$ and the x-axis will be touched at the point $(3, 0)$ . The coordinates for intersections are clear, with the intercepts being calculated as follows:

For $x = 0$ : $f(0) = |0 - 1| - 2 = 1 - 2 = -1 \text{, so } b = -1.$
For $x = 3$ : $f(3) = |3 - 1| - 2 = 2 - 2 = 0.$
Continuity points at $(1, -2)$ are important as well, confirming $a = -2$ as determined.

Step 2

b) $y = f(|x|)$

99%

104 rated

Answer

For the transformed function $y = f(|x|)$ , since absolute values affect symmetry, we can express the equation as follows:

For $|x| < 1$ , the function remains as $-x - 1$ .
For $|x| \geq 1$ , it is reflected hence: $f(|x|) = |x - 1| - 2$ will yield.
The graph will be symmetric about the y-axis, leading to intersections and confirming b values.

Intersections remain at symmetric points, thereby capturing key intercepts of the reflected graph.

Step 3

Given that $f(x) = |x - 1| - 2$, find (1) the value of $a$ and the value of $b$.

96%

101 rated

Answer

From the function $f(x) = |x - 1| - 2$ , we evaluate:

The vertex occurs at $(1, -2)$ , thus $a = -2$ .
From previous calculations, $b = -1$ from our earlier evaluation of $f(0)$ .

Step 4

Given that $f(x) = |x - 1| - 2$, find (2) the value of $x$ for which $f(x) = 5$.

98%

120 rated

Answer

To find $x$ for which $f(x) = 5$ , set up the equation:

$|x - 1| - 2 = 5$

This simplifies to:

$|x - 1| = 7$

This results in two equations to solve:

$x - 1 = 7$ leads to $x = 8$ .
$x - 1 = -7$ leads to $x = -6$ .

Thus, the solutions are $x = 8$ and $x = -6$ .

Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 2 - 2005 - Paper 5

Worked Solution & Example Answer:Figure 1 shows part of the graph of $y = f(x), x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 2 - 2005 - Paper 5

a) $y = f(x)$

b) $y = f(|x|)$

Given that $f(x) = |x - 1| - 2$, find (1) the value of $a$ and the value of $b$.

Given that $f(x) = |x - 1| - 2$, find (2) the value of $x$ for which $f(x) = 5$.

Join the A-Level students using SimpleStudy...