Polynomial Function
In a school contest, an observation drone is launched and the team records its height at each moment: $h(t) = -5t^2+20t+2$ (meters, with $t$ in seconds). That expression is a polynomial in $t$, and the graph of the flight is the graph of a polynomial function. What is the maximum height the drone reaches? In this section you formalize what you already know about lines and parabolas: domain, range, what happens when $x$ grows without bound, symmetry, and how to use roots to study the sign.
What a polynomial function is
Depending on the degree, each family has its own name: degree $1$ is a linear function $y=mx+b$ (a line), degree $2$ is a quadratic function (a parabola), and degree $3$ or higher is called a polynomial function of degree $n$. The range depends on the family: it is not always $\mathbb{R}$.
When several coefficients are missing, each point on the graph gives an equation: you substitute its coordinates into $f(x)=ax^2+bx+c$. With three points you get a system of three equations in $a$, $b$, and $c$.
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.
Given the quadratic function $f(x) = \frac{1}{2}x^2 - 4x + 1$, determine with justification its intervals of increase and decrease.
Given the quadratic function $f(x) = \frac{1}{2}x^2 - 4x + 1$, determine with justification its intervals of increase and decrease.
Parabola from its vertex
If you know the vertex $V(h, k)$, you don't need the system of three equations: use the vertex form and only one unknown remains, $a$.
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 $y = ax^2 + bx + c$.
The main cable of a suspension bridge takes the shape of a parabola. The horizontal distance between the support towers is $120$ meters, and the lowest point of the…
The main cable of a suspension bridge takes the shape of a parabola. The horizontal distance between the support towers is $120$ meters, and the lowest point of the cable is $20$ meters below the attachment point on the towers. Taking the lowest point of the cable as the origin of coordinates, find the equation of the parabola and calculate the height of the cable $30$ meters from the center.
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin of coordinates and that its vertex lies on…
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin of coordinates and that its vertex lies on the line $y = -x - 2$, find the analytical expression of the function or functions that satisfy these conditions.
End behavior and symmetry
The degree and the sign of the leading coefficient determine what happens to $f(x)$ as $x\to+\infty$ and as $x\to-\infty$.
Analyse the end behavior 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 behavior 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, factoring, 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 roots of the polynomial function $f(x) = x^3 - 6x^2 + 9x$ and study its sign across the real number line, stating the intervals where the function is positive…
Find the roots of the polynomial function $f(x) = x^3 - 6x^2 + 9x$ and study its sign across the real number line, stating the intervals where the function is positive and negative.
Polynomial from its roots and biquadratic equations
Roots also work in reverse: if you know them, you can build the polynomial. And when only even powers appear, a change of variable turns it into a quadratic.
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 $(0, 16)$.
Calculate the set of values of the parameter $k$ for which the polynomial function $P(x) = x^4 - 2x^2 + k$ has exactly two distinct real roots.
Calculate the set of values of the parameter $k$ for which the polynomial function $P(x) = x^4 - 2x^2 + k$ has exactly two distinct real roots.
Applications: optimizing
A very common use of the parabola is finding the maximum or minimum of a quantity: that value is at the vertex.
The height in meters 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 meters 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 $100$ meters 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 $100$ meters 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 maximize this area.
A projectile is launched upward such that its height in meters with respect to time in seconds follows the function $h(t) = -4.9t^2 + 19.6t + 1.5$. Calculate the maximum…
A projectile is launched upward such that its height in meters 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 beginning: with $h(t) = -5t^2+20t+2$, the vertex is at $t_v = -\dfrac{20}{2\cdot(-5)} = 2$ s and the maximum height is $h(2) = -5\cdot 2^2 + 20\cdot 2 + 2 = 22$ meters.
Revenue, profit, and volume models
Many situations are modeled by multiplying or subtracting simple functions, and the result is a polynomial.
The selling price in pesos of an item as a function of the number of units $x$ demanded on the market is given by the function $p(x) = 2\,400 - 40x$. Write the total…
The selling price in pesos of an item as a function of the number of units $x$ demanded on the market is given by the function $p(x) = 2\,400 - 40x$. Write the total revenue function $R(x)$ and calculate how many units must be sold to maximize revenue.
A technology company sells $x$ units of a certain electronic component. The unit selling price modeled in pesos is $p(x) = 4\,000 - 10x$. Total production costs follow…
A technology company sells $x$ units of a certain electronic component. The unit selling price modeled in pesos is $p(x) = 4\,000 - 10x$. Total production costs follow the function $C(x) = 100\,000 + 400x$. Express the profit function $B(x)$, determine the number of units that maximizes profit, and calculate this maximum profit.
A rectangular piece of cardboard measuring $40$ cm long by $30$ cm wide is available. An open-topped box is to be made by cutting out a square of side length $x$ from…
A rectangular piece of cardboard measuring $40$ cm long by $30$ cm wide is available. An open-topped box is to be made by cutting out a square of side length $x$ from each corner and folding up the resulting flaps. Express the volume $V(x)$ of the box as a polynomial function, state its valid domain in the context of this geometric problem, and calculate the volume if a square of side length $5$ cm is cut out.
Intersections and tangency
Calculate analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the straight line $y = 2x - 5$.
Calculate analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the straight line $y = 2x - 5$.
Determine the values of the parameter $m$ for which the line with equation $y = mx - 4$ is tangent to the parabola $y = x^2 - 3x + 5$.
Determine the values of the parameter $m$ for which the line with equation $y = mx - 4$ is tangent to the parabola $y = x^2 - 3x + 5$.