Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$. The curve has a turning point at $A(3, -4)$ and also passes through the point $(0, 5)$. (a) Write down the coordinates of the point to which A is transformed on the curve with equation (i) $y = |f(x)|$, (ii) $y = 2f(\frac{1}{2}x)$. (b) Sketch the curve with equation $y = f(|x|)$. On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis. The curve with equation $y = f(x)$ is a translation of the curve with equation $y = x^2$. (c) Find $f(x)$. (d) Explain why the function $f$ does not have an inverse.

A-Level Edexcel Maths Pure Rational Expressions: Figure 2 shows a sketch of the c

Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 5

Question 8

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Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$. The curve has a turning point at $A(3, -4)$ and also passes through the point... show full transcript

Worked Solution & Example Answer:Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 5

Step 1

Write down the coordinates of the point to which A is transformed on the curve with equation (i) $y = |f(x)|$

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Answer

For the point $A(3, -4)$ on the original curve, when transformed by $y = |f(x)|$ , the y-coordinate becomes non-negative. Thus, the transformed coordinates are $A'(3, 4)$ .

Step 2

Write down the coordinates of the point to which A is transformed on the curve with equation (ii) $y = 2f(\frac{1}{2}x)$

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Answer

To find the transformed coordinates for $y = 2f(\frac{1}{2}x)$ , we first substitute $x = 3$ to get $y = 2f(\frac{3}{2})$ . Since we don't have the value of $f(\frac{3}{2})$ , we express the transformed coordinates as $(3, 2f(\frac{3}{2}))$ .

Step 3

Sketch the curve with equation $y = f(|x|)$

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Answer

To sketch the curve $y = f(|x|)$ , reflect the portion of the curve for $x < 0$ across the y-axis, maintaining the turning points and the y-intercept. The point $(0, 5)$ remains the same, while the turning point $A(3, -4)$ becomes $A(-3, -4)$ .

Step 4

Find $f(x)$

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Answer

Since $y = f(x)$ is derived from the equation of a parabola $y = x^2$ , we can express $f(x)$ as:

$f(x) = x^2 - 7$

This aligns with the point $A(3, -4)$ since substituting $x = 3$ gives $f(3) = 3^2 - 7 = -4$ .

Step 5

Explain why the function $f$ does not have an inverse

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Answer

The function $f(x) = x^2 - 7$ is a parabolic function which opens upwards and possesses a turning point (minimum) at $x = 0$ . Since it is not one-to-one (it fails the horizontal line test), multiple $x$ values yield the same $y$ value. Therefore, $f$ does not have an inverse.

Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 5

Worked Solution & Example Answer:Figure 2 shows a sketch of the curve with the equation $y = f(x), \, x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 5

Write down the coordinates of the point to which A is transformed on the curve with equation (i) $y = |f(x)|$

Write down the coordinates of the point to which A is transformed on the curve with equation (ii) $y = 2f(\frac{1}{2}x)$

Sketch the curve with equation $y = f(|x|)$

Find $f(x)$

Explain why the function $f$ does not have an inverse

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