Polynomial function
A surveillance drone takes off and the control centre records its height at every instant: $h(t) = -5t^2+20t+2$. That expression is a polynomial in $t$, and the graph describing the drone's flight is the graph of a polynomial function. In this section you will formalise what you already know about straight lines and parabolas: domain, range, what happens when $x$ grows without bound, when there is symmetry, and how to use roots to study the sign of the function.
What a polynomial function is
Depending on the degree, each family has its own name and different properties: degree $1$ is a linear function (a straight line), degree $2$ is a quadratic function (a parabola), and degree $3$ or higher is simply called a polynomial function of degree $n$. The range, however, depends on the family: it is not always $\mathbb{R}$.
Calculate the value of the parameter $c$ so that the quadratic function $f(x) = 4x^2 - 3x + c$ passes through the point $(-1, 10)$.
Calculate the value of the parameter $c$ so that the quadratic function $f(x) = 4x^2 - 3x + c$ passes through the point $(-1, 10)$.
Determine the coefficients $a$, $b$ and $c$ of the quadratic polynomial function $f(x) = ax^2 + bx + c$, given that its graph passes through the points $(0, 3)$,…
Determine the coefficients $a$, $b$ and $c$ of the quadratic polynomial function $f(x) = ax^2 + bx + c$, given that its graph passes through the points $(0, 3)$, $(1, 4)$ and $(-1, 6)$.
Linear function: degree 1
Find the algebraic expression of the linear function passing through the points $A(2, -3)$ and $B(5, 6)$.
Find the algebraic expression of the linear function passing through the points $A(2, -3)$ and $B(5, 6)$.
Quadratic function: vertex and concavity
Determine the coordinates of the vertex and the equation of the axis of symmetry of the quadratic function $f(x) = 2x^2 - 8x + 5$.
Determine the coordinates of the vertex and the equation of the axis of symmetry of the quadratic function $f(x) = 2x^2 - 8x + 5$.
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ is concave (opening downwards) or convex (opening upwards) and calculate the exact coordinates…
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ is concave (opening downwards) or convex (opening upwards) and calculate the exact coordinates of its vertex.
Find the equation of the parabola whose vertex is the point $V(3, -2)$ and which passes through the point $P(5, 6)$. Express the result in the form $y = ax^2 + bx + c$.
Find the equation of the parabola whose vertex is the point $V(3, -2)$ and which passes through the point $P(5, 6)$. Express the result in the form $y = ax^2 + bx + c$.
End behaviour and symmetry
The degree and the sign of the leading coefficient determine what happens to $f(x)$ as $x\to+\infty$ and as $x\to-\infty$: the so-called end behaviour.
Analyse the end behaviour of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$ by determining the value of its limits as $x \to +\infty$ and as $x \to -\infty$.
Analyse the end behaviour of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$ by determining the value of its limits as $x \to +\infty$ and as $x \to -\infty$.
A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Given that it passes through the points $(1, -4)$ and $(2, 4)$, determine its algebraic expression and…
A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Given that it passes through the points $(1, -4)$ and $(2, 4)$, determine its algebraic expression and calculate all its axis intercepts.
Roots, factorisation and sign
Calculate analytically the axis intercepts of the polynomial function $p(x) = x^3 - 4x$.
Calculate analytically the axis intercepts of the polynomial function $p(x) = x^3 - 4x$.
Find the roots of the quadratic polynomial function $f(x) = x^2 - 5x + 6$.
Find the roots of the quadratic polynomial function $f(x) = x^2 - 5x + 6$.
Find the algebraic expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, given also that its graph intersects the y-axis at the…
Find the algebraic expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, given also that its graph intersects the y-axis at the point $(0, 16)$.
Applications: optimising and intersecting
Two very common uses of polynomial functions: finding the maximum or minimum of a quantity modelled by a parabola, and finding where two graphs intersect (two functions that take the same value at the same time).
The height in metres of an observation drone as a function of time in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Calculate the height of the drone at…
The height in metres of an observation drone as a function of time in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Calculate the height of the drone at times $t = 1$ and $t = 3$.
A farmer has $100$ metres of fencing to enclose a rectangular vegetable patch alongside a straight riverbank, so no fence is needed on that side. Express the area of the…
A farmer has $100$ metres of fencing to enclose a rectangular vegetable patch alongside a straight riverbank, so no fence is needed on that side. Express the area of the patch as a function of the width $x$ perpendicular to the river and determine the dimensions that maximise this area.
A projectile is launched upwards such that its height in metres with respect to time in seconds follows the function $h(t) = -4,9t^2 + 19,6t + 1,5$. Calculate the…
A projectile is launched upwards such that its height in metres with respect to time in seconds follows the function $h(t) = -4,9t^2 + 19,6t + 1,5$. Calculate the maximum height reached by the projectile and the time at which this occurs.
Back to the start: With the drone's function $h(t) = -5t^2+20t+2$, the vertex is reached at $t_v = -\dfrac{20}{2\cdot(-5)} = 2$ s, giving a maximum height of $h(2) = -5\cdot 2^2 + 20\cdot 2 + 2 = 22$ metres.