Transforming Graphs y=f(x) Worksheet with Answers

Transforming Graphs y=f(x)

discia.org 10 questions · 31 marks

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Either paper 4 marks

The diagram shows part of the graph of \(y = f(x)\).

The point \(P\), marked on the curve, has coordinates \((3, 1)\).

(a)
Write down the coordinates of the image of \(P\) on the graph of \(y = f(x) + 3\).

(b)
Write down the coordinates of the image of \(P\) on the graph of \(y = f(x + 3)\).

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(a) \((3, 4)\)
(b) \((0, 1)\)

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Either paper 4 marks

Here is part of the graph of \(y = f(x)\).

The point \(P\), marked on the curve, has coordinates \((5, 7)\).

(a)
Write down the coordinates of the image of \(P\) on the graph of \(y = -f(x)\).

(b)
Write down the coordinates of the image of \(P\) on the graph of \(y = f(-x)\).

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(a) \((5, -7)\)
(b) \((-5, 7)\)

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Either paper 3 marks

Here is part of the graph of \(y = f(x)\).

Its turning point \(T\) is at \((-1, -1)\).

(a)
Write down the coordinates of the turning point of the graph of \(y = f(x) - 4\).

(b)
Write down the coordinates of the turning point of the graph of \(y = f(x - 3)\).

(c)
Write down the coordinates of the turning point of the graph of \(y = -f(x)\).

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(a) \((-1, -5)\)
(b) \((2, -1)\)
(c) \((-1, 1)\)

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Either paper 2 marks

The graph of \(y = f(x)\) for \(-1 \le x \le 3\) is drawn on each grid below. It is made of straight lines joining five marked points.

(a)
On this grid, draw the graph of \(y = f(x) - 1\). Plot its five points, then join them.

(b)
On this grid, draw the graph of \(y = f(-x)\). Plot its five points, then join them.

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(a) The graph of \(y = f(x)\) moved down 1, the broken line through \((-1, -3)\), \((0, 0)\), \((1, 1)\), \((2, 2)\) and \((3, 0)\)
(b) The graph of \(y = f(x)\) reflected in the \(y\)-axis, the broken line through \((1, -2)\), \((0, 1)\), \((-1, 2)\), \((-2, 3)\) and \((-3, 1)\)
(a)

(b)

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Either paper 2 marks

Here is a sketch of \(y = f(x)\).

The curve passes through the points \((-2, -14)\), \((-1, -6)\), \((0, -4)\), \((1, -2)\) and \((2, 6)\).

On the grid, plot the five points that the curve \(y = f(x - 1)\) passes through in place of these, then draw the curve.

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A curve of the same shape through \((-1, -14)\), \((0, -6)\), \((1, -4)\), \((2, -2)\) and \((3, 6)\)

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Either paper 2 marks

Each grid shows a sketch of \(y = 3 \times 5^x\) as a dashed line.

(a)
Think about the graph of \(y = 3 \times 5^x + 2\).

On the grid, plot the point where it crosses the \(y\)-axis, and draw its asymptote.

(b)
Think about the graph of \(y = -3 \times 5^x\).

On the grid, plot the point where it crosses the \(y\)-axis, and draw its asymptote.

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(a) Crosses the \(y\)-axis at \((0, 5)\); asymptote \(y = 2\)
(b) Crosses the \(y\)-axis at \((0, -3)\); asymptote \(y = 0\)
(a)

(b)

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Either paper 3 marks

The solid curve is the graph of \(y = f(x)\). Its turning point is at \((2, -3)\).

The dashed curve is the image of the solid curve after a single transformation.

(a)
Describe fully the single transformation that maps the solid curve onto the dashed curve.

(b)
State the equation of the dashed curve in terms of \(f(x)\).

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(a) Translation by the vector \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\)
(b) \(y = f(x) + 3\)

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Either paper 3 marks

The solid curve is the graph of \(y = f(x)\). Its turning point is at \((1, -4)\).

The dashed curve is the image of the solid curve after a single transformation.

(a)
Describe fully the single transformation that maps the solid curve onto the dashed curve.

(b)
Write down the equation of the dashed curve in terms of \(f(x)\).

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(a) Translation by the vector \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\)
(b) \(y = f(x - 2)\)

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Either paper 4 marks

The solid curve is the graph of \(y = f(x)\). Its turning point is at \((1, -4)\).

The dashed curve is the image of the solid curve after a single transformation.

(a)
Describe fully the single transformation that maps the solid curve onto the dashed curve.

(b)
Write down the equation of the dashed curve in terms of \(f(x)\).

(c)
Write down the coordinates of the turning point of the dashed curve.

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(a) Reflection in the \(x\)-axis
(b) \(y = -f(x)\)
(c) \((1, 4)\)

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Either paper 4 marks

The graph of \(y = x^3 + 9\) is translated 5 units to the right.

The translated graph has equation \(y = f(x)\).

Calculate \(f(x)\).

Give your answer in the form \(x^3 + ax^2 + bx + d\) where \(a\), \(b\) and \(d\) are integers.

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\(f(x) = x^3 - 15x^2 + 75x - 116\)

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discia.org Transforming Graphs y=f(x) · sheet 1

Answers Transforming Graphs y=f(x) · sheet 1

(a) \((3, 4)\)
(b) \((0, 1)\)
(a) \((5, -7)\)
(b) \((-5, 7)\)
(a) \((-1, -5)\)
(b) \((2, -1)\)
(c) \((-1, 1)\)
(a) The graph of \(y = f(x)\) moved down 1, the broken line through \((-1, -3)\), \((0, 0)\), \((1, 1)\), \((2, 2)\) and \((3, 0)\)
(b) The graph of \(y = f(x)\) reflected in the \(y\)-axis, the broken line through \((1, -2)\), \((0, 1)\), \((-1, 2)\), \((-2, 3)\) and \((-3, 1)\)
(a)

(b)
A curve of the same shape through \((-1, -14)\), \((0, -6)\), \((1, -4)\), \((2, -2)\) and \((3, 6)\)
(a) Crosses the \(y\)-axis at \((0, 5)\); asymptote \(y = 2\)
(b) Crosses the \(y\)-axis at \((0, -3)\); asymptote \(y = 0\)
(a)

(b)
(a) Translation by the vector \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\)
(b) \(y = f(x) + 3\)
(a) Translation by the vector \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\)
(b) \(y = f(x - 2)\)
(a) Reflection in the \(x\)-axis
(b) \(y = -f(x)\)
(c) \((1, 4)\)
\(f(x) = x^3 - 15x^2 + 75x - 116\)

discia.org Transforming Graphs y=f(x) · sheet 1