Quadratic functions: parabolas

Knowing that the graph of the quadratic function $f(x) = ax^2 + c$ passes through the points $(0, 3)$ and $(2, 11)$, set up a system of equations and calculate the values of the parameters $a$ and $c$.

State which direction the branches of the parabola $y = -3x^2 + 6x - 1$ open, justifying your answer, and find out whether the vertex of this function corresponds to an absolute maximum or minimum. Calculate its coordinates.

Given the quadratic function $f(x) = 2x^2 - 8x + 5$, determine the exact coordinates of its vertex and the equation of its axis of symmetry.

Observe the three parabolas represented in the following figure. Match each of the algebraic expressions with its corresponding graph (A, B or C).

1) $y = -x^2 + 2$

2) $y = x^2 - 4x + 4$

3) $y = x^2 - 4$

During a training session, a volleyball player throws the ball upwards. The height $h$ (in metres) reached by the ball as a function of time $t$ (in seconds) is given by the expression $h(t) = -5t^2 + 10t + 2$. What is the maximum height reached by the ball and at what exact instant does it occur?

Calculate the points of intersection with the coordinate axes (both the x-axis and the y-axis) of the parabola $y = -x^2 + 2x + 8$.

Find algebraically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the line $y = -x + 3$.

Consider the family of parabolas given by the equation $y = x^2 - kx + 9$. Determine the values of the parameter $k$ for which the parabola is tangent to the x-axis (that is, it has a single point of intersection with the x-axis).

Determine the expanded algebraic expression of a quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(2, -1)$ and that it passes through the point $P(4, 3)$.

Look at the graph of the parabola. Determine the coordinates of its vertex and use an additional point from the graph to deduce its algebraic expression in the form $y = a(x-h)^2 + k$. Finally, expand it to the form $y = ax^2 + bx + c$.

Sketch the graph of the function $f(x) = |x^2 - 6x + 5|$. To do this, first find the vertex and the axes intercepts of the inner parabola without the modulus, and explain analytically how the modulus affects the final representation.

You can do it on paper and send a photo with your answer.

The school PTA is selling raffle tickets to fund the end-of-year trip. If each ticket is sold for $x$ euros, the estimated revenue (in tens of euros) is given by the quadratic function $I(x) = -2x^2 + 48x$. At what price should they sell the tickets to maximise revenue? What is the maximum revenue?

A farmer wants to build a rectangular pen for his chickens by taking advantage of an existing long stone wall, so that he only needs to fence three of the four sides of the rectangle. He has $100$ metres of wire fencing available. Determine the dimensions of the pen so that its area is as large as possible, and calculate this maximum area.

Given the parabolas $y = x^2 - 2x - 3$ and $y = -x^2 + 4x + 5$, find analytically the exact points where they intersect. Then calculate the straight-line distance between the vertices of both parabolas.