True or False? The correct term for a multiplicative transformation is 'dilation', whether the graph gets bigger or smaller.

True. Stretch and compression are useful for picturing which of the two is happening, but dilation is the correct term for both. A full description gives the word dilation together with the correct dilation factor.

True or False? The horizontal and vertical transformations in a combined transformation can be applied in either order.

True. The two groups are independent of one another, so doing all the horizontal ones first gives the same result as doing all the vertical ones first. It is only within a group that the order makes a difference.

When a graph is transformed, what are the preimage and the image ?

The preimage is the set of points before the transformations are applied, that is the points on the graph of the original function. The image is the set of points afterwards, on the graph of the transformed function.

Which transformations affect the domain, and which affect the range?

Horizontal transformations, both translations and dilations, affect the domain . Vertical transformations, both translations and dilations, affect the range .

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Transformations of Functions (College Board AP® Precalculus): Flashcards

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FrontVertical & Horizontal Translations
Define additive transformation.

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Cards in this collection (33)

Define additive transformation.
An additive transformation adds a constant either to the output of a function, as in $f (x) + k$ , or to the input, as in $f (x + h)$ .
Both kinds produce a translation of the graph, shifting it without changing its shape, size or orientation.
What does $g (x) = f (x) + k$ do to the graph of $f$ ?
It translates the graph vertically by $k$ units, up when $k > 0$ and down by $|k|$ when $k < 0$ .
Every point moves by the same amount, so the shape of the graph is unchanged.
True or False?
The graph of $f (x + 3)$ is the graph of $f$ moved $3$ units to the right.
False.
Adding a positive constant inside the function moves the graph in the opposite direction, so $f (x + 3)$ shifts $3$ units to the left.
It is $f (x - 2)$ that shifts to the right, by $2$ units.
How do you tell a vertical translation from a horizontal one?
Ask whether the constant is applied outside or inside the function.
A constant added outside, to $f (x)$ , changes the output and gives a vertical shift, while one added inside, to $x$ , changes which input produces each output and gives a horizontal shift.
Which features of a graph are unchanged by a vertical translation?
Anything determined by the input values: the vertical asymptotes, the intervals of increase and decrease, and the concavity intervals.
The $x$ values of local maxima and minima also stay put, even though their $y$ values shift by $k$ .
The function $f$ is given by $f (x) = x^{2} - 3 x + 4$ . Fill in the two gaps to find $g (x) = f (x) + 2$ .
$g (x) = x^{2} - 3 x + 4 +______= x^{2} - 3 x +______$
The completed working is:
$g (x) = x^{2} - 3 x + 4 + 2 = x^{2} - 3 x + 6$
Adding a constant outside the function simply adds it to the whole expression.
Which features of a graph are unchanged by a horizontal translation?
Anything determined by the output values, which in practice means the horizontal asymptotes.
The $y$ values of local maxima and minima also stay the same, even though the $x$ values at which they occur all shift.
True or False?
A vertical translation generally moves the zeros of a function.
True.
Every output value changes by $k$ , so an input that previously gave an output of $0$ usually no longer does.
The $y$ -intercept shifts by $k$ as well, and a horizontal asymptote at $y = 0$ becomes $y = k$ .
How do you find the analytical form of $f (x + h)$ ?
Replace $x$ everywhere in the expression for $f (x)$ with $x + h$ , then simplify.
For $f (x) = x^{2} - 3 x + 4$ this gives $f (x + 2) = {(x + 2)}^{2} - 3 (x + 2) + 4 = x^{2} + x + 2$ .
Define multiplicative transformation.
A multiplicative transformation multiplies either the output of a function, as in $a f (x)$ , or the input, as in $f (b x)$ .
Both kinds produce a dilation of the graph, moving every point towards or away from one of the axes.
What does $g (x) = a f (x)$ do to the graph of $f$ ?
It dilates the graph vertically by a factor of $|a|$ , moving every point that many times as far from the $x$ -axis.
It is a stretch when $|a| > 1$ and a compression when $0 < |a| < 1$ .
By what factor does $g (x) = f (b x)$ dilate the graph horizontally?
By a factor of $\frac{1}{|b|}$ , which is the reciprocal of the constant rather than the constant itself.
So multiplying the input by a number greater than $1$ compresses the graph towards the $y$ -axis instead of stretching it.
True or False?
A negative value of $a$ in $a f (x)$ only changes the size of the dilation.
False.
The dilation factor is $|a|$ , but a negative $a$ also reflects the graph over the $x$ -axis.
In the same way a negative $b$ in $f (b x)$ reflects the graph over the $y$ -axis.
Which features of a graph are unchanged by a vertical dilation?
The zeros stay in the same place, because $a$ multiplied by $0$ is still $0$ , and so do the vertical asymptotes.
The $y$ -intercept, the $y$ values of local extrema and any horizontal asymptote are all multiplied by $a$ .
The function $f$ is given by $f (x) = x^{2} - 3 x + 4$ . Fill in the two gaps to find $g (x) = 2 f (x)$ .
$g (x) = 2 (x^{2} - 3 x + 4) =______x^{2} - 6 x +______$
The completed working is:
$g (x) = 2 (x^{2} - 3 x + 4) = 2 x^{2} - 6 x + 8$
Every term is multiplied by $a$ , because it is the whole function that is being multiplied.
Which features of a graph are unchanged by a horizontal dilation?
The $y$ -intercept stays the same, since $f (b \cdot 0) = f (0)$ , and horizontal asymptotes stay at the same $y$ value.
The $x$ values of the zeros, of the local extrema and of any vertical asymptotes are all multiplied by $\frac{1}{b}$ .
True or False?
The correct term for a multiplicative transformation is 'dilation', whether the graph gets bigger or smaller.
True.
Stretch and compression are useful for picturing which of the two is happening, but dilation is the correct term for both.
A full description gives the word dilation together with the correct dilation factor.
How do you find the analytical form of $f (b x)$ ?
Replace $x$ everywhere in the expression for $f (x)$ with $b x$ , then simplify.
For $f (x) = x^{2} - 3 x + 4$ this gives $f (2 x) = {(2 x)}^{2} - 3 (2 x) + 4 = 4 x^{2} - 6 x + 4$ .
Complete the factored form of a combined transformation of $f$ .
$g (x) = a f (b (x +______)) +______$
The completed form is:
$g (x) = a f (b (x + h)) + k$
Writing a combined transformation this way lets each constant be read off as the single transformation it causes.
In $a f (b (x + h)) + k$ , which constants cause horizontal changes and which cause vertical ones?
The constants inside the function, $b$ and $h$ , cause the horizontal dilation and the horizontal translation.
The constants outside, $a$ and $k$ , cause the vertical dilation and the vertical translation.
In what order are the transformations in $a f (b (x + h)) + k$ applied?
Within each group, dilations and reflections come first, then translations.
So from $f (x)$ the horizontal steps run to $f (b x)$ and then on to $f (b (x + h))$ .
True or False?
The horizontal and vertical transformations in a combined transformation can be applied in either order.
True.
The two groups are independent of one another, so doing all the horizontal ones first gives the same result as doing all the vertical ones first.
It is only within a group that the order makes a difference.
When a graph is transformed, what are the preimage and the image?
The preimage is the set of points before the transformations are applied, that is the points on the graph of the original function.
The image is the set of points afterwards, on the graph of the transformed function.
How do you write $g (x) = - f (2 x + 3) - 1$ in factored form?
Factor the constant multiplying $x$ out of the whole input expression, giving $g (x) = - f (2 (x + 1.5)) - 1$ .
That has $a = - 1$ , $b = 2$ , $h = 1.5$ and $k = - 1$ .
The function $g$ is given by $g (x) = f (3 (x + 5))$ . What sequence of transformations maps $f$ to $g$ ?
A horizontal dilation by a factor of $\frac{1}{3}$ , followed by a horizontal translation of $5$ units to the left.
The dilation comes first because the expression is already in factored form, where dilations precede translations.
True or False?
A combined transformation whose translations happen before its dilations cannot be written in factored form.
False.
Any such transformation can be rewritten in factored form, which then describes the same end result with the dilations first.
Translating $5$ right and then dilating horizontally by a factor of $2$ with a reflection gives $f (- \frac{1}{2} (x + 10))$ , a dilation followed by a translation $10$ units left.
Which transformations affect the domain, and which affect the range?
Horizontal transformations, both translations and dilations, affect the domain.
Vertical transformations, both translations and dilations, affect the range.
The function $f$ has range $[r, s]$ . What is the range of $f (x) + k$ ?
It is $[r + k, s + k]$ , the original range shifted by $k$ .
The domain is unchanged, because the inputs to the function have not been altered.
The function $f$ has domain $[p, q]$ . What is the domain of $f (x + h)$ ?
It is $[p - h, q - h]$ , the original domain shifted by $- h$ .
The range is unchanged, because the outputs of the function have not been altered.
True or False?
If $f$ has range $[r, s]$ and $a < 0$ , then $a f (x)$ has range $[a r, a s]$ .
False.
Multiplying by a negative number reverses the inequality, so the two endpoints swap and the range is $[a s, a r]$ .
The same reversal happens to the domain when $b < 0$ in a horizontal dilation.
When does a horizontal transformation leave the domain of a function completely unchanged?
When the domain is already all real numbers, since translating or dilating an unbounded interval horizontally leaves it unbounded.
Every polynomial has domain all real numbers, so transformations matter most for functions with restricted domains such as $f (x) = \sqrt{x}$ .
The function $f$ is given by $f (x) = \frac{1}{x - 2}$ , so its domain excludes $x = 2$ . Fill in the two gaps for $g (x) = f (x - 3)$ .
Substituting gives $g (x) = \frac{1}{x -______}$ so the excluded value has moved to $x =______$ instead.
The completed working is:
Substituting gives $g (x) = \frac{1}{x - 5}$ so the excluded value has moved to $x = 5$ instead.
The domain is still unbounded, but the excluded value has shifted $3$ to the right along with the graph.
The function $f$ has domain $[- 2, 6]$ and range $[1, 9]$ . What are the domain and range of $2 f (x - 4) + 3$ ?
The domain is $[2, 10]$ , since only the horizontal shift of $4$ to the right affects it.
The range is $[5, 21]$ , from scaling $[1, 9]$ by $2$ to give $[2, 18]$ and then shifting up by $3$ .

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