Inverse Variation and the Hyperbola y = k/x
Aaliyah and her friends are splitting the $\$60$ cost of a pizza party. With 2 people, each pays $\$30$. With 3 people, each pays $\$20$. With 6 people, each pays $\$10$. The more people, the less each person pays, but exactly how does it work? And what would it even mean for $0$ people to split the bill?
Inverse variation
In the pizza party, the number of people $x$ times the share per person $y$ is always the same: $2 \cdot 30 = 60$, $3 \cdot 20 = 60$, $6 \cdot 10 = 60$. When a product stays constant, one quantity goes up exactly as the other goes down.
Why $y = k/x$? If you double $x$, then $y$ must be cut in half so that the product $xy$ stays the same. Compare this with direct variation $y = kx$, where doubling $x$ doubles $y$.
The curve above is the pizza party: the share per person $y = \dfrac{60}{x}$ for $x$ people.
Identify in the following table whether the quantities $x$ and $y$ are inversely proportional and, if so, find the constant of proportionality $k$ and write the function…
Identify in the following table whether the quantities $x$ and $y$ are inversely proportional and, if so, find the constant of proportionality $k$ and write the function that relates them.
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$x$: 1, 2, 3, 4, 6
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$y$: 24, 12, 8, 6, 4
Given the inverse proportion function $y = \frac{18}{x}$, calculate the value of $y$ for $x = 3$, $x = -9$ and $x = 0,5$.
Given the inverse proportion function $y = \frac{18}{x}$, calculate the value of $y$ for $x = 3$, $x = -9$ and $x = 0,5$.
The graph: hyperbola, domain, range and asymptotes
Two facts control the whole graph of $y = \dfrac{k}{x}$. First, you can never divide by $0$. Second, a fraction with a nonzero numerator is never $0$.
The graph is a hyperbola with two separate pieces called branches. It gets closer and closer to the axes without ever touching them.
Where the branches sit depends on the sign of $k$. Try the slider.
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If $k > 0$, the branches are in quadrants I and III.
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If $k < 0$, the branches are in quadrants II and IV.
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The larger $|k|$ is, the farther the curve sits from the axes.
Determine the domain, the range and the equations of the asymptotes of the function $y = \frac{-8}{x}$.
Determine the domain, the range and the equations of the asymptotes of the function $y = \frac{-8}{x}$.
Match each of the following functions with the description of the quadrants in which their branches are located when graphed: a) $y = \frac{5}{x}$ b) $y = \frac{-2}{x}$…
Match each of the following functions with the description of the quadrants in which their branches are located when graphed:
- a) $y = \frac{5}{x}$
- b) $y = \frac{-2}{x}$
- c) $y = \frac{0,1}{x}$
Translated hyperbolas: y = k/(x − h) + v
What if the graph is shifted? Replacing $x$ with $x - h$ slides the graph $h$ units to the right. Adding $v$ slides it $v$ units up. The asymptotes slide along with it.
Why the range excludes $v$: the fraction $\dfrac{k}{x-h}$ is never $0$, so $y = \dfrac{k}{x-h} + v$ is never $0 + v$.
State the equations of the vertical and horizontal asymptotes of the hyperbola whose algebraic expression is $y = \frac{5}{x-4} + 3$.
State the equations of the vertical and horizontal asymptotes of the hyperbola whose algebraic expression is $y = \frac{5}{x-4} + 3$.
Find the domain of the rational function $y = \frac{7}{x+5}$.
Find the domain of the rational function $y = \frac{7}{x+5}$.
The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units down. Write the algebraic expression of the new…
The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units down. Write the algebraic expression of the new function, determine its domain and find the coordinates of the point where its asymptotes intersect.
Intercepts and increasing/decreasing
To find intercepts of $y = \dfrac{k}{x-h} + v$, use the same idea as for any function.
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$y$-intercept: substitute $x = 0$ (possible only if $h \neq 0$).
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$x$-intercept: set $y = 0$ and solve. Isolate the fraction first, then solve the equation.
A translated hyperbola can have one $x$-intercept and one $y$-intercept, or only one of them. For example, $y = \dfrac{3}{x - 2}$ has horizontal asymptote $y = 0$, so it never reaches the $x$-axis.
For $y = \dfrac{6}{x-3} - 2$, we have $k = 6 > 0$, so $y$ is decreasing for $x < 3$ and decreasing for $x > 3$.
Find analytically the points of intersection with the coordinate axes of the rational function $y = \frac{10}{x-2} + 5$.
Find analytically the points of intersection with the coordinate axes of the rational function $y = \frac{10}{x-2} + 5$.
Determine the domain, the range, the asymptotes and the points of intersection with the axes of the function $y = \frac{-3}{x+2} - 4$. Also state its intervals of…
Determine the domain, the range, the asymptotes and the points of intersection with the axes of the function $y = \frac{-3}{x+2} - 4$. Also state its intervals of increase and decrease.
Reading an equation from a graph
Every hyperbola of this type is determined by two things: its asymptotes (which fix $h$ and $v$) and one point (which fixes $k$).
For a hyperbola centered at the origin, steps 1–2 are trivial and $k = xy$ for any point on it.
Look at the following graph of an inverse proportion function and determine its constant $k$ and its algebraic expression $y = \frac{k}{x}$.
Look at the following graph of an inverse proportion function and determine its constant $k$ and its algebraic expression $y = \frac{k}{x}$.
From the graph of this translated hyperbola, determine the equations of its asymptotes, the coordinates of its center of symmetry and deduce its algebraic expression in…
From the graph of this translated hyperbola, determine the equations of its asymptotes, the coordinates of its center of symmetry and deduce its algebraic expression in the form $y = \frac{k}{x-a} + b$.
Building an equation from information
The same four steps work when you are given words instead of a graph. Each piece of information is either an asymptote (it gives $h$ or $v$), the center (it gives both), or a point on the curve (it gives $k$). An intercept is a point too.
Write the equation of a hyperbola of the form $y = \frac{k}{x-a} + b$ knowing that its asymptotes are the lines $x = -2$ and $y = 4$, and that its graph passes through…
Write the equation of a hyperbola of the form $y = \frac{k}{x-a} + b$ knowing that its asymptotes are the lines $x = -2$ and $y = 4$, and that its graph passes through the point $(0, 5)$.
A hyperbola of the form $y = \frac{k}{x-a} + b$ has its center of symmetry at the point $(-1, 3)$ and intersects the x-axis at $x = 1$. Find its algebraic expression and…
A hyperbola of the form $y = \frac{k}{x-a} + b$ has its center of symmetry at the point $(-1, 3)$ and intersects the x-axis at $x = 1$. Find its algebraic expression and calculate the point of intersection with the y-axis.
Rewriting a rational function
Sometimes a hyperbola shows up as $y = \dfrac{2x + 7}{x + 3}$, with $x$ in both the numerator and denominator. To find its asymptotes, rewrite it in the form $\dfrac{k}{x - h} + v$. The trick: write the numerator using the denominator.
Some problems call this polynomial division: dividing $2x + 7$ by $x + 3$ gives quotient $2$ and remainder $1$, so $\dfrac{2x+7}{x+3} = 2 + \dfrac{1}{x+3}$. The next example shows a quick way to get there.
The two curves coincide: they are the same function.
Express the rational function $y = \frac{2x+7}{x+3}$ in the form $y = \frac{k}{x-a} + b$ by performing polynomial division. Then, state the equations of its asymptotes…
Express the rational function $y = \frac{2x+7}{x+3}$ in the form $y = \frac{k}{x-a} + b$ by performing polynomial division. Then, state the equations of its asymptotes and the coordinates of its center of symmetry.
Hyperbola meets line
To find where a hyperbola and a line intersect, set the two expressions equal and solve. Multiply both sides by the denominator to clear the fraction; you will get a quadratic equation.
How many intersections can there be? After clearing the fraction you have a quadratic, so the discriminant $b^2 - 4ac$ tells you: positive means two points, zero means one point (the line is tangent: it touches the curve at exactly one point), and negative means none.
Find analytically the points of intersection between the hyperbola $y = \frac{-4}{x-1}$ and the line $y = -x + 4$.
Find analytically the points of intersection between the hyperbola $y = \frac{-4}{x-1}$ and the line $y = -x + 4$.
Find the value of the constant $k$ so that the hyperbola $y = \frac{k}{x}$ is tangent to the line $y = -x + 6$ (that is, so that they intersect at a single point).
Find the value of the constant $k$ so that the hyperbola $y = \frac{k}{x}$ is tangent to the line $y = -x + 6$ (that is, so that they intersect at a single point).
Modeling with inverse variation
Inverse variation appears whenever a total is shared or a job is split: more people means less each, more speed means less time. The key step is to find the quantity that stays constant, because that is $k$.
A lottery prize of $\$300$ is to be shared equally among a group of $x…
A lottery prize of $\$300$ is to be shared equally among a group of $x$ friends. Write the algebraic expression of the function that relates the money each one receives to the number of friends and calculate how much they receive if there are $6$ people.
To travel a fixed distance of $120$ miles, the time $t$ taken (in hours) and the average speed $v$ (in mph) are inversely proportional quantities. Write the function…
To travel a fixed distance of $120$ miles, the time $t$ taken (in hours) and the average speed $v$ (in mph) are inversely proportional quantities. Write the function $v(t)$ and calculate at what constant speed a car must travel if it wants to take exactly $2,5$ hours to complete the trip.
To fill a swimming pool, $4$ faucets with the same flow rate take $9$ hours. Write the function that relates the filling time $t$ (in hours) to the number of open…
To fill a swimming pool, $4$ faucets with the same flow rate take $9$ hours. Write the function that relates the filling time $t$ (in hours) to the number of open faucets $n$. How long would $6$ faucets take to fill the same pool?
Science models, inequalities and formulas
In science, inverse variation hides in formulas. Boyle's law says that for a gas at constant temperature, pressure $P$ and volume $V$ satisfy $PV = k$.
Often you need a condition such as "at most" or "at least," which gives an inequality. If the variable is positive (as speed, area, volume or pressure are), you can multiply both sides by it without flipping the inequality sign.
Another common situation is a formula that has the unknown in two reciprocals. Solve for the variable you want, then rewrite the result as a hyperbola.
According to Boyle's law, at a constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \cdot V = k$). If a given gas…
According to Boyle's law, at a constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \cdot V = k$). If a given gas occupies a volume of $20$ liters at a pressure of $1,5$ atmospheres, write the function $V(P)$ and calculate what volume it will occupy at $3$ atmospheres.
The electrical resistance $R$ (in ohms, $\Omega$) of a conducting wire of constant length is inversely proportional to its cross-sectional area $S$ (in mm$^2$). If for a…
The electrical resistance $R$ (in ohms, $\Omega$) of a conducting wire of constant length is inversely proportional to its cross-sectional area $S$ (in mm$^2$). If for a cross-section of $2 \text{ mm}^2$ the resistance is $0,02\ \Omega$, express $R$ as a function of $S$. What must the minimum cross-section of the wire be so that the resistance is less than or equal to $0,01\ \Omega$?
The focal length of a thin converging lens is $20$ inches. The relationship between the object distance to the lens $s_o$ and the image distance $s_i$ is given by the…
The focal length of a thin converging lens is $20$ inches. The relationship between the object distance to the lens $s_o$ and the image distance $s_i$ is given by the equation $\frac{1}{20} = \frac{1}{s_o} + \frac{1}{s_i}$. Express $s_i$ as a function of $s_o$. What happens to the image distance when the object approaches $20$ inches from the lens from the right?
Cost models and the domain in context
In a real situation, the math domain ($x \neq 0$) is not always the logical domain. If $x$ is a number of people, it must be a positive whole number.
Rational functions also model average cost: total cost divided by the number of items. A food truck has $\$300$ in fixed daily expenses and spends $\$3$ per meal.
A group of people rents a tennis court for $\$24$. Write the function $C(x)$ that expresses the cost per person as a function of the number of players $x…
A group of people rents a tennis court for $\$24$. Write the function $C(x)$ that expresses the cost per person as a function of the number of players $x$. If the final cost was $\$6$ per person, how many players rented the court? What is the logical domain of this function taking into account the context of the problem?
An electronic components factory has fixed daily expenses of $\$500$ to turn on the machinery, and producing each unit costs $\$2$. Write the function for the average…
An electronic components factory has fixed daily expenses of $\$500$ to turn on the machinery, and producing each unit costs $\$2$. Write the function for the average cost per unit $C_m(x)$ as a function of the number of units $x$ manufactured per day. Find its asymptotes and interpret the meaning of the horizontal asymptote in the context of the problem.
Summary
Back to the start: Aaliyah's pizza party is $y = \dfrac{60}{x}$. The product $xy = 60$ stays constant, the branch in quadrant I describes the real situation (only positive whole numbers of people), and $x = 0$ is excluded: the vertical asymptote says that the fewer the people, the bigger the share each one pays.