The graph of the curve with equation $y = f(x)$ is shown on the grid below. (a) On the grid above, sketch the graph of the curve with equation $y = f(-x)$. The curve $C$ with equation $y = 5 + 2x - x^2$ is transformed by a translation to give the curve $S$ such that the point $(1, 6)$ on $C$ is mapped to the point $(4, 6)$ on $S$. (b) Find an equation for $S$.

GCSE Edexcel Maths Coordinate Geometry: The graph of the curve with equa

The graph of the curve with equation $y = f(x)$ is shown on the grid below - Edexcel - GCSE Maths - Question 21 - 2020 - Paper 2

Question 21

The graph of the curve with equation $y = f(x)$ is shown on the grid below. (a) On the grid above, sketch the graph of the curve with equation $y = f(-x)$. The cur... show full transcript

Worked Solution & Example Answer:The graph of the curve with equation $y = f(x)$ is shown on the grid below - Edexcel - GCSE Maths - Question 21 - 2020 - Paper 2

Step 1

On the grid above, sketch the graph of the curve with equation $y = f(-x)$

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Answer

To sketch the graph of $y = f(-x)$ , we reflect the graph of $y = f(x)$ across the y-axis. This involves taking each point $(x, y)$ on the original graph and transforming it to $(-x, y)$ .

Thus, the critical points of the original function should be mirrored:

The point $(0, 2)$ becomes $(0, 2)$ (remains unchanged).
The point $(2, 0)$ becomes $(-2, 0)$ .
The point $(4, 0)$ becomes $(-4, 0)$ .

Following this transformation, sketch the curve ensuring it passes through the newly reflected points.

Step 2

Find an equation for $S$

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Answer

To find the equation for curve $S$ , we first determine the translation that occurs from curve $C$ to curve $S$ . The point $(1, 6)$ on curve $C$ is mapped to $(4, 6)$ on curve $S$ .

The horizontal translation can be calculated as $4 - 1 = 3$ . As there is no change in the y-coordinate, the vertical translation is $0$ .

Thus, we apply a horizontal shift of 3 units to the left for all x-coordinates in the original equation of curve $C$ .

The equation for curve $C$ is: $y = 5 + 2x - x^2$

After applying the translation, substitute $x$ with $(x - 3)$ in the equation: $y = 5 + 2(x - 3) - (x - 3)^2$

Expanding this gives: $y = 5 + 2x - 6 - (x^2 - 6x + 9)$ $y = 5 - 6 + 2x + 6x - 9$ $y = 8x - x^2 - 10$

Thus, the equation for curve $S$ is: $y = -x^2 + 8x - 10$

The graph of the curve with equation $y = f(x)$ is shown on the grid below - Edexcel - GCSE Maths - Question 21 - 2020 - Paper 2

Worked Solution & Example Answer:The graph of the curve with equation $y = f(x)$ is shown on the grid below - Edexcel - GCSE Maths - Question 21 - 2020 - Paper 2

On the grid above, sketch the graph of the curve with equation $y = f(-x)$

Find an equation for $S$

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