Exercises: Second-degree polynomials: Parabolas

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.

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

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) $y = x^2 - 4$

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.

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

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

Cian has to find the axis of symmetry and the range of the parabola $f(x) = -x^2 + 6x - 5$. On his exam, he writes the following:

"Since $a = -1$ and $b = 6$, the $x$-coordinate of the vertex is $x_v = \frac{-b}{2a} = \frac{-6}{2 \times (-1)} = 3$. Therefore, the axis of symmetry is the horizontal line $y = 3$. Next, I calculate the $y$-coordinate of the vertex: $y_v = f(3) = -(3)^2 + 6 \times 3 - 5 = 4$, so the vertex is $V(3, 4)$. Since $a < 0$, the branches open downward and the parabola has a maximum, so its range is $[4, +\infty)$".

1. a) Identify the two conceptual errors Cian made in his answer.
2. b) Explain with reasoning why they are incorrect, referring to the relationship between the axis of symmetry and vertical lines, and to the graphical meaning of the range when the branches open downward.
3. c) Write the corrected solution: the correct equation of the axis of symmetry and the interval defining the range of $f(x)$.

The design of the parabolic arch of an ornamental fountain follows the quadratic function $f(x) = ax^2 + bx + c$. The water jet starts from ground level at the point $(0, 0)$ and touches the ground again exactly at a horizontal distance of $6\text{ metres}$, at the point $(6, 0)$. In addition, it is measured that the maximum height reached by the jet is $4.5\text{ metres}$.

1. a) Without previously calculating the algebraic coefficients, deduce with reasoning the equation of the vertical line that acts as the axis of symmetry and the exact coordinates of its vertex $V(h, k)$.
2. b) Use the vertex form $f(x) = a(x-h)^2 + k$ or the factored form $f(x) = a(x-x_1)(x-x_2)$ to determine the numerical value of parameter $a$.
3. c) Write the expanded equation $f(x) = ax^2 + bx + c$ and verify that both the ground contact points and the maximum height reached are satisfied.

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?

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?

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

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

We have $40$ metres 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?

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

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 $(-1, -2)$.

To graph the parabola $g(x) = 2x^2 - 12x + 10$, two classmates debate which strategy is faster and more reliable to determine its vertex and its intercepts with the axes:

• Cian proposes factoring out the leading coefficient to write $g(x) = 2(x^2 - 6x + 5)$, factoring the polynomial inside the parentheses by finding two integers with product $5$ and sum $-6$, and obtaining the vertex using the fact that the axis of symmetry is equidistant from both real roots.

• Aoife argues that it is more direct to transform the function into its vertex form $g(x) = a(x-h)^2 + k$ by completing the square from $2(x^2 - 6x) + 10$.

1. a) Fully carry out both procedures step by step until finding the coordinates of the vertex $V(h, k)$ and confirm that both lead to the same solution.
2. b) Compare both methods and justify which one is more advantageous in this specific case. In what situation would the method proposed by Cian become unfeasible or extremely cumbersome compared to Aoife's method or the general formula $x_v = \frac{-b}{2a}$? Support your reasoning with the role of the discriminant $\Delta$.

Met Éireann, the Irish Meteorological Service, models the ambient temperature $T(t)$, in degrees Celsius ($^\circ\text{C}$), during the central hours of a summer day in Dublin as a function of the time of day $t$ (with $t \in [8, 24]$, where $t=8$ corresponds to 8:00 h and $t=24$ to 24:00 h) using the quadratic function:

1. a) Determine with reasoning at what time of day the maximum temperature is recorded and what that extreme value is.
2. b) Set up and solve the necessary inequality to calculate in which time interval the temperature was greater than or equal to $40\ ^\circ\text{C}$.
3. c) Justify analytically by calculating the discriminant $\Delta$ of the equation $T(t) = 48$ whether the model predicts that $48\ ^\circ\text{C}$ will be reached or exceeded at any time of day.

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.

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 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.

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).

The school robotics club sells sensor kits to fund its materials. It has monthly fixed overheads of $2000$ euros and assembling each kit costs an additional $15$ euros. The monthly revenue obtained from selling $x$ kits is given by the function $I(x) = -0.1x^2 + 65x$. Write the function representing the monthly profit $B(x)$. How many kits must they assemble and sell to obtain maximum profit?

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

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 $18$.

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 $(5, -1)$.

The municipal transport company of an Irish city (Cork) analyzes its fares based on urban mobility data. With the current ticket price at $€1.20$, an average of $6000$ people travel daily. A technical study estimates that for every $€0.10$ increase in the fare, $250$ daily passengers will be lost (and vice versa if it is decreased). The daily revenue function is expressed as $I(x) = (1.20 + 0.10x)(6000 - 250x)$, where $x$ is the number of $€0.10$ increments.

1. a) Estimate before calculating: If it is decided to double the ticket price (becoming $€2.40$, which means $x = 12$), do you expect daily revenue to be greater than, less than, or equal to the initial $€7200$? Justify your prediction qualitatively by weighing the price increase against the loss of users.
2. b) Calculate: Expand the algebraic expression of $I(x)$ into standard form $I(x) = ax^2 + bx + c$, find the coordinates of its vertex, and deduce the number of increments $x$ that maximizes revenue, the resulting optimal ticket price, and the maximum daily revenue.
3. c) Check and reflect: Calculate the exact value of $I(12)$ to quantitatively check your estimate from part a) and interpret the result using the axis of symmetry of the parabola. Finally, reflect critically: from the perspective of public service and urban sustainability, why is it not always advisable to set the fare at the optimal vertex of the parabola if this entails a price increase?

Consider the family of quadratic functions given by $f_m(x) = (m-1)x^2 + 2mx + (m+3)$, where $m$ is a real parameter with $m \neq 1$.

1. a) Determine with reasoning for which values of $m$ the graph of $f_m(x)$ has branches that open upward (reaching an absolute minimum) and, at the same time, does not intersect the horizontal axis $X$ at any point. Express the solution set as an interval.
2. b) Knowing that one of the parabolas in this family has the vertical line $x = 2$ as its axis of symmetry, find the specific value of $m$, write the expanded equation of said parabola, and deduce its exact range, justifying whether the function is bounded above or below.
3. c) Prove algebraically whether there is any value of $m$ for which the vertex of the parabola lies exactly on the $X$-axis. If so, calculate the complete coordinates of that vertex; if not, explain why it is impossible.