Quadratic Functions and Parabolas

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

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.

Determine, by substituting into the equation, whether the point $(-1, 4)$ and the point $(2, 1)$ lie on the graph of the parabola $y = 2x^2 - 3x - 1$.

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) $y = x^2 - 4$

The function $d(t) = 5t^2$ describes the distance in centimeters traveled by a marble rolling down an inclined track as a function of time $t$ in seconds. What distance will it have traveled after $4$ seconds? How long will it take to travel exactly $125$ cm?

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 $c$.

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).

Express the function $y = (x + 3)^2 - 2$ as the result of applying a translation to the basic parabola $y = x^2$. Clearly state the direction and number of units translated along both the horizontal and vertical axes.

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 $y = ax^2 + bx + c$.

Find the analytical expression of the quadratic function $f(x) = ax^2 + bx + c$ knowing that it intersects the x-axis at the points $x = -1$ and $x = 5$, and that the maximum value reached by the function is $18$.

Calculate the coefficients $a$, $b$ and $c$ of the quadratic function $y = ax^2 + bx + c$ knowing that its graph passes through the points $(0, -5)$, $(1, -6)$ and $(-1, -2)$.

An arch of a stone bridge is in the shape of a parabola. The base of the arch (at water level) is $12$ meters wide and the maximum height at the center is $6$ meters. If we place the origin of coordinates at the center of the base of the arch, find the equation of the parabola that describes its profile.

A basketball player shoots at the basket from a height of $2$ meters. The ball follows a parabolic trajectory, reaching its maximum height of $4$ meters at a horizontal distance of $2$ meters from the point of release. Knowing that the rim of the basket is $3$ meters high and at a horizontal distance of $4.5$ meters, deduce the equation of the trajectory and show analytically whether or not the ball will go through the rim.

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.

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.

Given the following piecewise function:

$f(x) = \begin{cases} -x^2 + 4x & \text{if } x \le 3 \\ mx + n & \text{if } x > 3 \end{cases}$

Determine the values of the parameters $m$ and $n$ so that the function is continuous at the boundary point $x = 3$ and, furthermore, the half-line passes through the point $(5, -1)$.

During a training session, a volleyball player throws the ball upward. The height $h$ (in meters) 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?

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

We have $40$ meters of rope to enclose a rectangular play area at a camp. If we let $x$ be the length of the base of the rectangle, write the function that determines the area of the enclosure in terms of $x$. What should the dimensions of the rectangle be so that the enclosed area is as large as possible?

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 $100$ meters of wire fencing available. Determine the dimensions of the pen so that its area is as large as possible, and calculate this maximum area.

The school robotics club sells sensor kits to fund its materials. It has monthly fixed costs of $40\,000$ pesos and assembling each kit costs an additional $300$ pesos. The monthly revenue obtained from selling $x$ kits is given by the function $I(x) = -2x^2 + 1\,300x$. Write the function representing the monthly profit $B(x)$. How many kits must they assemble and sell to obtain maximum profit?