Exercises: Quadratic function (parabola)

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 points of intersection with the coordinate axes of the quadratic function $y = -x^2 + x + 6$.

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.

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?

Determine the domain and range of the function $f(x) = -3x^2 + 12x - 1$.

Write the equation of the parabola obtained by shifting the graph of $y = -2x^2$ three units to the right and five units upwards.

Calculate the value of the coefficient $c$ so that the parabola with equation $y = x^2 - 6x + c$ passes through the point $P(2, -3)$.

State the intervals where the function $f(x) = \frac{1}{2}x^2 + 3x - 2$ is increasing and decreasing.

TYPE: SOMEONE ELSE'S ERROR. Marcos is analyzing the economic viability of a small artisanal footwear workshop in Elche (Alicante). The function that estimates the monthly net profit (in hundreds of euros) depending on the price $x$ (in euros) set for each pair of espadrilles is $B(x) = -2x^2 + 80x - 600$.

In his report, Marcos writes the following:

“Since $a = -2$ and $b = 80$, the $x$-coordinate of the vertex is $x_v = \frac{-b}{2a} = \frac{-80}{-4} = 20$. Therefore, the optimal price is $20\text{ €}$. However, since the coefficient $a$ is negative, the parabola opens downwards, which means that at $20\text{ €}$ we obtain the minimum possible profit; therefore, we should never sell at that price.”

1. a) Point out the conceptual error Marcos makes when interpreting the direction of the branches and the vertex when $a < 0$. Clearly explain why his conclusion is false.
2. b) Correct the report by calculating the actual profit obtained by selling at $20\text{ €}$ (remember that the result is in hundreds of euros) and indicate whether this is a maximum or a minimum.
3. c) Without performing long calculations, explain what relationship there will be between the profit obtained by setting a price of $15\text{ €}$ and that obtained by setting a price of $25\text{ €}$, based on the function's axis of symmetry.

TYPE: MODELING. During a heatwave episode in Córdoba, the State Meteorological Agency (AEMET) records that the temperature $T(t)$ in $^{\circ}\text{C}$ during daytime hours (between $8:00$ and $20:00$) fits the quadratic function with remarkable precision:

where $t$ is the time elapsed in hours since $8:00$ in the morning (that is, $t = 0$ corresponds to $8:00$ and $t = 12$ to $20:00$).

1. a) At what real time of day (official clock time) is the maximum temperature reached, and what is that value according to the model?
2. b) Check whether the model predicts the same temperature at $10:00$ ($t = 2$) as at $18:00$ ($t = 10$). Justify why this happens using the concept of axis of symmetry without calculating both temperatures separately.
3. c) AEMET activates an orange alert for heat in the Córdoba countryside when the temperature reaches or exceeds $40\text{ }^{\circ}\text{C}$. Based on the range of the function in this interval, reason whether the orange alert threshold was reached on this day.

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?

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 $y = 2$.

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 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:

Determine the coefficients $a$, $b$ and $c$ of the quadratic function $f(x) = ax^2 + bx + c$, knowing that its graph passes through the points $A(0, 3)$, $B(1, 4)$ and $C(-1, 6)$.

The monthly profit of a company, in thousands of euros, is given by the function $B(x) = -2x^2 + 120x - 1000$, where $x$ is the number of units sold (in hundreds). How many units must it sell to maximise its profit, and how much is this profit? Between which values of $x$ does the company not make a loss?

Write the algebraic expression of a quadratic function that has an absolute maximum of $8$ at $x = 3$ and whose graph passes through the origin.

Given the function $f(x) = -2(x - 3)(x + 1)$, determine, without expanding it to polynomial form, its x-intercepts, the x-coordinate of its vertex, and its maximum value.

TYPE: CHOOSE THE METHOD. A technical staff of a basketball club in the Liga Endesa analyzes the parabola traced by the ball during a free throw. To study the height $y$ (in meters) with respect to the horizontal distance $x$ (in meters from the free-throw line), two analysts present the same function written in two different forms:

• Analyst A (standard form): $f(x) = -0,2x^2 + 1,2x + 2$

• Analyst B (vertex form): $g(x) = -0,2(x - 3)^2 + 3,8$

The coaching staff needs to answer two immediate questions:

1. What is the maximum height reached by the ball, and at what horizontal distance from the shooter does it occur?

2. At what exact height does the ball leave the player's hands at the moment of the shot ($x = 0$)?

1. a) To answer question 1, which of the two expressions is more advantageous and direct? Justify your answer by indicating how both values are obtained without having to apply additional formulas.
2. b) To answer question 2, which expression allows you to answer immediately and why?
3. c) Prove that both expressions are equivalent by algebraically expanding Analyst B's formula.

TYPE: MODELING / ESTIMATE AND CHECK. Red Eléctrica de España (REE) monitors the production of a photovoltaic solar farm in Extremadura during a clear spring day. The power generated $P(t)$ (in megawatts, $\text{MW}$) between sunrise ($6:00$) and sunset ($18:00$) is modeled by the function:

where $t$ is the time in hours elapsed since $6:00$ ($0 \le t \le 12$).

1. a) Estimate: Knowing that at $6:00$ ($t=0$) and at $18:00$ ($t=12$) the power is zero ($0\text{ MW}$), estimate at what time of day you would expect solar radiation and production to be at their maximum by natural symmetry. What approximate power do you intuitively expect the plant to reach?
2. b) Check: Analytically calculate the coordinates of the vertex of the parabola. Does the exact calculated time match your previous estimate? What is the actual maximum power generated?
3. c) At $14:00$ ($t = 8$), the plant's telematics system records a generation of $480\text{ MW}$. An operator suspects that there is an anomaly in the sensors. Evaluate $P(8)$ using the model and determine whether the sensor reading is correct or if there really is a technical fault.
4. d) Determine the range of the function on the considered domain and interpret what the obtained minimum value means in the context of the electrical supply.

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.

Given the parabolas $f(x) = x^2 - 4x + 3$ and $g(x) = -x^2 + bx + c$, find the values of $b$ and $c$ such that both graphs intersect at the vertex of $f(x)$ and on the y-axis.

Sketch the graph of the function $f(x) = |x^2 - 6x + 5|$ after first expressing it as a piecewise-defined function. Determine its intervals of increase and decrease and its local extrema.

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

In a fireworks display, a rocket is launched from a platform $10\text{ m}$ high. It follows a parabolic trajectory, reaching a height of $25\text{ m}$ at a horizontal distance of $20\text{ m}$, and explodes upon hitting the ground at a horizontal distance of $60\text{ m}$. Determine the equation of the trajectory $y = ax^2 + bx + c$.

Calculate the value of the parameter $k$ so that the line with equation $y = 2x + k$ is tangent to the parabola $y = x^2 - 4x + 5$. Also determine the coordinates of the point of tangency.

A theatre has a capacity of $500$ people. With the ticket price at $20\text{ \euro}$, the theatre is completely full. A market study indicates that for every euro the ticket price increases, $10$ fewer spectators will attend. At what price should the ticket be set to maximise the theatre's revenue? What will the maximum revenue be?

Determine the coefficients $a$ and $b$ of the function $f(x) = ax^2 + bx - 3$, knowing that it has a local minimum at $x = -2$ and that the average rate of change in the interval $[-2, 1]$ is equal to $3$.

Consider the family of parabolas given by the equation $f_m(x) = mx^2 - 2mx + 1$, where $m$ is a non-zero real parameter. Show that all parabolas in this family pass through two fixed points regardless of the value of $m$, and calculate the coordinates of these points.

TYPE: INVERSE PROBLEM. In the construction project of a bypass road in Asturias, the Demarcación de Carreteras must design the entrance of a cut-and-cover tunnel with a symmetrical parabolic profile to protect the road from rockfalls. The cross-section is modeled with a parabola of the form $f(x) = a(x - h)^2 + k$, where $x$ represents the horizontal position in meters measured from the center of the road (that is, the center is at $x = 0$) and $y = f(x)$ is the clearance height in meters.

The technical requirements set out in the specifications are:

1. The base of the tunnel on the ground must have a total width of $12\text{ meters}$ (resting on the ground at $x = -6$ and $x = 6$, where the height is $0\text{ m}$).

2. The maximum height of the tunnel must be located at the center of the road and be exactly $6\text{ meters}$.

3. Road safety regulations: At $4\text{ meters}$ from the center (at $x = 4$ and $x = -4$, where trucks travel along the outer lanes), the tunnel clearance must be at least $3\text{ meters}$.

1. a) Identify with reasoning the coordinates of the vertex $(h, k)$ from the data and state the equation in vertex form.
2. b) Use the ground contact point ($x = 6$, $y = 0$) to calculate the exact value of parameter $a$. Write the complete quadratic function that defines the tunnel profile.
3. c) Check analytically whether the design meets road safety regulations by calculating the exact height at $x = 4$. Will heavy vehicles be able to pass safely?

TYPE: MODELING. An olive oil cooperative in Jaén sells $5\text{-liter}$ carafes of extra virgin olive oil. Based on sales records from the previous season, the governing board has the following data:

• At the base price of $30\text{ €}$ per carafe, they sell an average of $1200\text{ carafes}$ per month.

• For each $2\text{ €}$ increase applied to the price, demand decreases by $50\text{ carafes}$ per month due to customers switching to other brands.

Let $x$ be the number of $2\text{ €}$ increases applied (so that the price per carafe is $P(x) = 30 + 2x$ and the quantity sold is $V(x) = 1200 - 50x$).

1. a) Obtain the algebraic expression for the cooperative's monthly revenue function, $I(x) = P(x) \cdot V(x)$, in its expanded form $ax^2 + bx + c$. Justify, from the sign of the quadratic coefficient, why this mathematical relationship ensures the existence of a maximum revenue.
2. b) Determine how many $2\text{ €}$ increases must be applied to maximize monthly revenue, and deduce what the final retail price of each carafe should be.
3. c) The commercial director proposes: “To be successful, we must sell as much product as possible; raising the price reduces sales and always harms the cooperative.” Mathematically refute his claim by comparing the revenue obtained by selling at the base price of $30\text{ €}$ versus that obtained with the calculated optimal price.
4. d) Starting from how many $2\text{ €}$ increases would the cooperative have no customers left? State the realistic domain of the variable $x$ and the range of the revenue function in this commercial context.