Quadratic function (parabola)
A ball thrown towards a basketball hoop, the jet of water from a fountain or the cable of a suspension bridge all trace the same curve: a parabola. Unlike the straight lines you already know, here $x$ is squared, and that changes everything: the graph has a highest or lowest point, and it does not always increase (or decrease) at the same rate.
The parabola and its vertex
Determine the coordinates of the vertex, the equation of the axis of symmetry and the direction of the arms of the parabola $f(x) = 2x^2 - 8x + 5$.
Determine the coordinates of the vertex, the equation of the axis of symmetry and the direction of the arms of the parabola $f(x) = 2x^2 - 8x + 5$.
Calculate the value of the coefficient $c$ so that the parabola with equation $y = x^2 - 6x + c$ passes through the point $P(2, -3)$.
Calculate the value of the coefficient $c$ so that the parabola with equation $y = x^2 - 6x + c$ passes through the point $P(2, -3)$.
Intercepts with the axes
As with any function: the intercept with the $y$-axis is found by calculating $f(0)$ (there is always one). For the $x$-axis we solve the equation $ax^2+bx+c=0$; there can be two, one or no intercepts, depending on the sign of the discriminant.
Calculate the points of intersection with the coordinate axes of the quadratic function $y = -x^2 + x + 6$.
Calculate the points of intersection with the coordinate axes of the quadratic function $y = -x^2 + x + 6$.
Calculate the average rate of change of the function $f(x) = x^2 - 4x$ in the intervals $[1, 3]$ and $[3, 5]$. What do the signs obtained indicate about whether the…
Calculate the average rate of change of the function $f(x) = x^2 - 4x$ in the intervals $[1, 3]$ and $[3, 5]$. What do the signs obtained indicate about whether the function is increasing or decreasing in those intervals?
Domain, range and rate of change
The domain of any quadratic function is $\mathbb{R}$: there is no forbidden $x$. The range depends on the vertex and on which way the branches open.
Determine the domain and range of the function $f(x) = -3x^2 + 12x - 1$.
Determine the domain and range of the function $f(x) = -3x^2 + 12x - 1$.
State the intervals where the function $f(x) = \frac{1}{2}x^2 + 3x - 2$ is increasing and decreasing.
State the intervals where the function $f(x) = \frac{1}{2}x^2 + 3x - 2$ is increasing and decreasing.
Translations of the parabola
Observe the graph of the following parabola, which is a translation of $y = x^2$. Write its algebraic expression in the form $f(x) = (x - h)^2 + k$ and expand it to…
Observe the graph of the following parabola, which is a translation of $y = x^2$. Write its algebraic expression in the form $f(x) = (x - h)^2 + k$ and expand it to express it as a quadratic polynomial.
Write the equation of the parabola obtained by shifting the graph of $y = -2x^2$ three units to the right and five units upwards.
Write the equation of the parabola obtained by shifting the graph of $y = -2x^2$ three units to the right and five units upwards.
Find the equation of the parabola in the form $y = ax^2 + bx + c$, knowing that its vertex is the point $V(-1, 4)$ and that it intercepts the y-axis at $y = 2$.
Find the equation of the parabola in the form $y = ax^2 + bx + c$, knowing that its vertex is the point $V(-1, 4)$ and that it intercepts the y-axis at $y = 2$.
Trajectories and optimisation
Many physical phenomena (throws, falling objects) and "maximum profit" or "maximum area" problems are modelled with a parabola: the answer is almost always at its vertex.
The throw of a javelin follows a parabolic trajectory given by the function $h(x) = -0,05x^2 + 2x + 1,5$, where $x$ is the horizontal distance in metres from the thrower…
The throw of a javelin follows a parabolic trajectory given by the function $h(x) = -0,05x^2 + 2x + 1,5$, where $x$ is the horizontal distance in metres from the thrower and $h(x)$ is the height in metres. What is the maximum height reached by the javelin? At what distance from the thrower does it hit the ground?
Analytically determine the intersection points between the parabola $f(x) = x^2 - 5x + 4$ and the line $g(x) = -x + 1$.
Analytically determine the intersection points between the parabola $f(x) = x^2 - 5x + 4$ and the line $g(x) = -x + 1$.
A farmer has $120\text{ m}$ of fencing to enclose a rectangular field adjacent to a straight river. If he does not need to put fencing along the river bank, what should…
A farmer has $120\text{ m}$ of fencing to enclose a rectangular field adjacent to a straight river. If he does not need to put fencing along the river bank, what should the dimensions of the field be for its area to be maximum? What is this maximum area?
The diagram illustrates the layout of the field and the river:
We want to inscribe a rectangle in an isosceles triangle of base $8\text{ cm}$ and height $6\text{ cm}$, with one of the sides of the rectangle resting on the base of…
We want to inscribe a rectangle in an isosceles triangle of base $8\text{ cm}$ and height $6\text{ cm}$, with one of the sides of the rectangle resting on the base of the triangle. Express the area of the rectangle as a function of half its base, $x$, and determine the dimensions of the rectangle with maximum area.