Function transformations
Two hot-air balloons at a festival in Kilkenny descend with the same flight profile, but one starts ten minutes later: its graph is the same, simply shifted. Recognising these "same drawing, moved or stretched" cases saves us from having to study the function from scratch every time.
Translations: moving the graph without distorting it
Given the function $f(x) = x^2 - 3x$, write the algebraic expression for the function $g(x)$ obtained by translating the graph of $f(x)$ four units to the left and two…
Given the function $f(x) = x^2 - 3x$, write the algebraic expression for the function $g(x)$ obtained by translating the graph of $f(x)$ four units to the left and two units upwards.
A hot-air balloon descends at a constant speed. Its altitude in metres is given by $A(t) = 800 - 50t$, where $t$ is the time in minutes. A second balloon begins its…
A hot-air balloon descends at a constant speed. Its altitude in metres is given by $A(t) = 800 - 50t$, where $t$ is the time in minutes. A second balloon begins its descent $10$ minutes later with exactly the same flight profile. Write the expression for the altitude of the second balloon $B(t)$ as a function of $A(t)$ and expand its final algebraic expression.
Observe the graph of the function $f(x)$ defined on the interval $[-4, 4]$. Determine the domain and range of the transformed function $g(x) = f(x-3) + 1$.
Observe the graph of the function $f(x)$ defined on the interval $[-4, 4]$. Determine the domain and range of the transformed function $g(x) = f(x-3) + 1$.
Domain and range of a transformed function
Translations, dilations and reflections change the domain and range in the same way they move the points. Horizontal changes ($x$-values) affect the domain; vertical changes ($y$-values) affect the range. In $g(x)=a\,f(b(x-h))+k$ the domain endpoints are mapped by $x \to \frac{x}{b}+h$ and the range endpoints by $y \to ay+k$. If $b$ or $a$ is negative, the endpoints swap order.
Dilations and reflections
The graph of a function $f(x)$ passes through the point $P(-3, 5)$. Determine the coordinates of the corresponding point on the graphs of the following transformed…
The graph of a function $f(x)$ passes through the point $P(-3, 5)$. Determine the coordinates of the corresponding point on the graphs of the following transformed functions:
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$g(x) = -f(x)$
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$h(x) = f(-x)$
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$i(x) = 2f(x)$
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$j(x) = f(x) - 4$
Given the basic function $f(x) = \frac{1}{x}$, state which geometric transformations (translations, stretches, reflections) must be applied successively, and in what…
Given the basic function $f(x) = \frac{1}{x}$, state which geometric transformations (translations, stretches, reflections) must be applied successively, and in what order, to obtain the function $g(x) = -\frac{3}{x} + 2$.
The function $C(x)$ represents the cost in euros of producing $x$ electronic components. Due to a technological improvement on the assembly line, three times as many…
The function $C(x)$ represents the cost in euros of producing $x$ electronic components. Due to a technological improvement on the assembly line, three times as many components can now be produced for exactly the same total cost. Express the new cost function $N(x)$ in terms of the original function $C(x)$.
Combining several transformations: order matters
Rational functions and changes of axes
A function such as $f(x)=\frac{2x+1}{x-1}$ is a transformation of $y=\frac{1}{x}$ in disguise. Divide first: $\frac{2x+1}{x-1}=2+\frac{3}{x-1}$. So the graph is $y=\frac1x$ stretched vertically by factor $3$, shifted $1$ unit right and $2$ units up. The centre of the hyperbola, and its centre of symmetry, is $(1, 2)$: the asymptotes are $x=1$ and $y=2$.
Changing the axes is also a transformation. If the new origin is at $(p, q)$ of the old system, then new coordinates are $X=x-p$ and $Y=y-q$. Substitute $x=X+p$ and $y=Y+q$ into the old equation. If the new $X$ direction is reversed, use $x=p-X$. If the unit on the $Y$ axis is halved, then $y=\frac{Y}{2}+q$ in the new measure.
The graph of the function $f(x)$ consists of two line segments joining the points $(-4, -2)$, $(0, 4)$ and $(4, -2)$. A new function is defined by the transformation…
The graph of the function $f(x)$ consists of two line segments joining the points $(-4, -2)$, $(0, 4)$ and $(4, -2)$. A new function is defined by the transformation $g(x) = -f\left(\frac{x}{2}\right) + 3$.
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Describe the ordered sequence of geometric transformations applied to $f(x)$.
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Calculate the exact coordinates of the new vertices of the graph of $g(x)$.
Given the basic function $f(x) = \sqrt{x}$, we want to obtain the function $g(x) = \sqrt{4x - 12} + 5$. Explain in detail two different sequences of geometric…
Given the basic function $f(x) = \sqrt{x}$, we want to obtain the function $g(x) = \sqrt{4x - 12} + 5$.
Explain in detail two different sequences of geometric transformations (altering the order of horizontal translations and stretches) that allow obtaining $g(x)$ from $f(x)$. Prove algebraically that both sequences lead to the same expression.
Express the rational function $f(x) = \frac{2x + 7}{x + 1}$ in the form $f(x) = k + \frac{m}{x+1}$ by carrying out polynomial division. From this result, describe step…
Express the rational function $f(x) = \frac{2x + 7}{x + 1}$ in the form $f(x) = k + \frac{m}{x+1}$ by carrying out polynomial division. From this result, describe step by step the geometric transformations that must be applied to the rectangular hyperbola $y = \frac{1}{x}$ to obtain the graph of $f(x)$.
Find the values of the parameters $a$ and $b$ so that the graph of the rational function $f(x) = \frac{ax + 3}{x + b}$ is symmetrical about the point $C(4, 2)$.
Find the values of the parameters $a$ and $b$ so that the graph of the rational function $f(x) = \frac{ax + 3}{x + b}$ is symmetrical about the point $C(4, 2)$.
Transformations with absolute value
For $g(x)=|f(x)+m|$, first shift $f$ vertically by $m$, then reflect the part below the $X$ axis. The number of $X$-intercepts of $g$ equals the number of solutions of $f(x)=-m$. For a hyperbola with horizontal asymptote $y=c$, the graph never reaches the value $c$, so $f(x)=-m$ has no solution when $-m=c$.
From the graph of the function $f(x)$ shown in the figure, describe the geometric steps required to obtain the graph of the function $g(x) = |f(x)|$ and calculate the…
From the graph of the function $f(x)$ shown in the figure, describe the geometric steps required to obtain the graph of the function $g(x) = |f(x)|$ and calculate the coordinates of the new vertices or points of inflection.
Express the function $g(x) = |-x^2 + 4|$ as a piecewise function by removing the absolute value and clearly stating the intervals where the original expression changes…
Express the function $g(x) = |-x^2 + 4|$ as a piecewise function by removing the absolute value and clearly stating the intervals where the original expression changes sign.
Match each of the algebraic expressions with the corresponding panel in the figure, given that all of them derive from transformations involving absolute values applied…
Match each of the algebraic expressions with the corresponding panel in the figure, given that all of them derive from transformations involving absolute values applied to the base function $y = x - 3$.
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$y = |x| - 3$
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$y = ||x-3| - 2|$
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$y = ||x| - 3|$
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$y = |x-3| - 2$
The figure shows the light curve of a variable star, modelled by the function $L(t) = 4 \sin\left(\frac{\pi}{6}t\right) + 10$, where $t$ is measured in days and $L$ in…
The figure shows the light curve of a variable star, modelled by the function $L(t) = 4 \sin\left(\frac{\pi}{6}t\right) + 10$, where $t$ is measured in days and $L$ in units of luminosity. A dense cloud of cosmic dust obscures the star such that the luminosity $O(t)$ observed from Earth is exactly the absolute value of the difference between $L(t)$ and its mean luminosity.
Determine the algebraic expression for $O(t)$, calculate its new period and find at which times $t \in [0, 12]$ the observed luminosity is maximum.
Back to the start: If the height of the first balloon as a function of time $t$ (in minutes) is $h(t)$, the height of the second balloon, which sets off ten minutes later, is described exactly as $g(t) = h(t - 10)$, a horizontal translation of $10$ units to the right.