Power functions
The area of a circle is $A(r) = \pi r^2$, the volume of a cube is $V(a) = a^3$ and the intensity of light decreases as $I(d) = k\cdot d^{-2}$. In all three cases, the variable is in the base and the exponent is fixed: these are power functions. Depending on whether that exponent is positive, negative or fractional, the shape of the graph changes dramatically.
What a power function is
Identify which of the following algebraic expressions correspond to a power function: $f(x) = 3x^2$ $g(x) = 2^x$ $h(x) = \frac{5}{x^3}$ $i(x) = x^{1/2}$
Identify which of the following algebraic expressions correspond to a power function:
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$f(x) = 3x^2$
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$g(x) = 2^x$
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$h(x) = \frac{5}{x^3}$
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$i(x) = x^{1/2}$
Write the algebraic expression for the function that assigns each positive real number its cube root, expressing it in index form with a fractional exponent.
Write the algebraic expression for the function that assigns each positive real number its cube root, expressing it in index form with a fractional exponent.
Calculating images and finding the coefficient
To find $f(a)$, substitute $x=a$ and work out the power. If you know a point on the graph, you can solve for the coefficient $a$ in $f(x)=a\cdot x^n$.
Given the power function $f(x) = 2x^4$, calculate the value of its outputs for $x = -2$, $x = 0$ and $x = 3$.
Given the power function $f(x) = 2x^4$, calculate the value of its outputs for $x = -2$, $x = 0$ and $x = 3$.
Find the value of the coefficient $k$ of the power function $f(x) = k \cdot x^3$ knowing that its graph passes through the point with coordinates $(2, 40)$.
Find the value of the coefficient $k$ of the power function $f(x) = k \cdot x^3$ knowing that its graph passes through the point with coordinates $(2, 40)$.
Knowing that the graph of a power function of the form $f(x) = x^n$ passes through the point $(3, 81)$, find the value of the exponent $n$.
Knowing that the graph of a power function of the form $f(x) = x^n$ passes through the point $(3, 81)$, find the value of the exponent $n$.
Negative exponent: two asymptotes
State the domain of definition and the equations of the vertical and horizontal asymptotes of the power function $f(x) = x^{-2}$.
State the domain of definition and the equations of the vertical and horizontal asymptotes of the power function $f(x) = x^{-2}$.
Given the function $f(x) = -3x^{-3}$, determine its domain of definition, its range, and study algebraically whether it exhibits any type of symmetry (even or odd).
Given the function $f(x) = -3x^{-3}$, determine its domain of definition, its range, and study algebraically whether it exhibits any type of symmetry (even or odd).
Fractional exponent: the curve that starts at the origin
A fractional exponent $x^{p/q}$ is equivalent to a root: $x^{1/2}=\sqrt{x}$, $x^{2/3}=\sqrt[3]{x^2}$. If the denominator $q$ is even, the domain is restricted to $x\geq0$.
State the domain of definition and determine the type of symmetry (even, odd or neither) of the power function $f(x) = x^6$.
State the domain of definition and determine the type of symmetry (even, odd or neither) of the power function $f(x) = x^6$.
Given the power function with a fractional exponent $f(x) = x^{2/3}$: Determine its domain of definition. Calculate the exact values of $f(8)$ and $f(-8)$. State with…
Given the power function with a fractional exponent $f(x) = x^{2/3}$:
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Determine its domain of definition.
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Calculate the exact values of $f(8)$ and $f(-8)$.
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State with algebraic reasoning whether the function is even, odd or neither.
Power functions in science
Many physical and biological laws have a power form: area as a function of radius, kinetic energy as a function of speed, Kepler's third law.
Express the area $A$ of a circle as a power function in terms of its radius $r$. What is the degree of this power function?
Express the area $A$ of a circle as a power function in terms of its radius $r$. What is the degree of this power function?
The kinetic energy of a moving body is given by the formula $E_c(v) = \frac{1}{2} m v^2$. If the mass of a vehicle is $1200$ kg, express its kinetic energy as a power…
The kinetic energy of a moving body is given by the formula $E_c(v) = \frac{1}{2} m v^2$. If the mass of a vehicle is $1200$ kg, express its kinetic energy as a power function dependent on its speed $v$ (in m/s) and calculate the energy it has when travelling at $20$ m/s.
The volume of a sphere in terms of its radius is calculated using the formula $V(r) = \frac{4}{3}\pi r^3$. Rearrange for the radius $r$ to express it as a power function…
The volume of a sphere in terms of its radius is calculated using the formula $V(r) = \frac{4}{3}\pi r^3$. Rearrange for the radius $r$ to express it as a power function dependent on the volume $V$. Use this new function to calculate the approximate radius of a sphere that has a volume of $1000$ cm$^3$ (take $\pi \approx 3,14$).
The basal metabolic rate $T$ (in kcal/day) of mammals is modelled in terms of their body mass $M$ (in kg) using Kleiber's power law: $T(M) = 70 \cdot M^{3/4}$. Calculate…
The basal metabolic rate $T$ (in kcal/day) of mammals is modelled in terms of their body mass $M$ (in kg) using Kleiber's power law: $T(M) = 70 \cdot M^{3/4}$. Calculate the basal metabolic rate of a $16$ kg dog. Then, find the approximate mass of an animal whose metabolic rate is $7000$ kcal/day.
Kepler's third law states that the square of the orbital period $T$ of a planet is directly proportional to the cube of its mean distance to the Sun $R$, that is,…
Kepler's third law states that the square of the orbital period $T$ of a planet is directly proportional to the cube of its mean distance to the Sun $R$, that is, $T^2 = k \cdot R^3$. If for Earth $T = 1$ year and $R = 1$ AU (Astronomical Unit), express the period $T$ as a power function dependent on $R$. Use this function to calculate the orbital period of Jupiter, knowing that its mean distance to the Sun is $5,2$ AU.