Exercises: Polynomial function

Determine the coordinates of the vertex and the equation of the axis of symmetry of the quadratic function $f(x) = 2x^2 - 8x + 5$.

Find the algebraic expression of the linear function passing through the points $A(2, -3)$ and $B(5, 6)$.

Calculate analytically the axis intercepts of the polynomial function $p(x) = x^3 - 4x$.

The height in metres of an observation drone as a function of time in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Calculate the height of the drone at times $t = 1$ and $t = 3$.

State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ is concave (opening downwards) or convex (opening upwards) and calculate the exact coordinates of its vertex.

Analyse the end behaviour of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$ by determining the value of its limits as $x \to +\infty$ and as $x \to -\infty$.

Find the roots of the quadratic polynomial function $f(x) = x^2 - 5x + 6$.

Calculate the value of the parameter $c$ so that the quadratic function $f(x) = 4x^2 - 3x + c$ passes through the point $(-1, 10)$.

Marcos wants to determine the range of the quadratic polynomial function $f(x) = -2x^2 + 4x + 1$. To do this, he calculates the x-coordinate of the vertex using the formula $x_v = -\dfrac{b}{2a} = -\dfrac{4}{2 \cdot (-2)} = 1$, finds its y-coordinate by calculating $f(1) = -2(1)^2 + 4(1) + 1 = 3$, and concludes on his exam: "Since the vertex is at $(1, 3)$, the function starts at $y = 3$ and goes up indefinitely, so its range is $[3, +\infty)$ ".\n\na) Point out the conceptual mistake Marcos makes and explain what property of the parabola he has overlooked.\nb) Correct his reasoning by justifying the curvature (concavity or convexity) from the sign of the leading coefficient.\nc) Write down the correct range of the function.

At an AEMET weather station in the province of Teruel, the temperature $T$ (in degrees Celsius) during a freezing early morning is modeled by the quadratic function $T(t) = 0,5t^2 - 4t + 3$, where $t$ represents the time in hours elapsed since midnight ($0 \le t \le 8$).\n\na) Before performing any algebraic calculation, estimate with reasoning from the signs and magnitudes of the coefficients whether the minimum temperature will be reached at the beginning of the night (before 2:00) or towards the middle of the period, and whether that temperature will become negative.\nb) Calculate the exact coordinates of the vertex of the parabola to find the time at which the minimum value is recorded and the exact temperature reached.\nc) Compare the analytical result obtained with your initial estimate and assess whether the model reflects a plausible meteorological behavior for that time range.

Find the equation of the parabola whose vertex is the point $V(3, -2)$ and which passes through the point $P(5, 6)$. Express the result in the form $y = ax^2 + bx + c$.

A farmer has $100$ metres of fencing to enclose a rectangular vegetable patch alongside a straight riverbank, so no fence is needed on that side. Express the area of the patch as a function of the width $x$ perpendicular to the river and determine the dimensions that maximise this area.

A projectile is launched upwards such that its height in metres with respect to time in seconds follows the function $h(t) = -4,9t^2 + 19,6t + 1,5$. Calculate the maximum height reached by the projectile and the time at which this occurs.

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

The selling price in euros of an item as a function of the number of units $x$ demanded on the market is given by the function $p(x) = 120 - 2x$. Write the total revenue function $R(x)$ and calculate how many units must be sold to maximise revenue.

Calculate analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the straight line $y = 2x - 5$.

Given the quadratic function $f(x) = \frac{1}{2}x^2 - 4x + 1$, determine with justification its intervals of increase and decrease.

Find the roots of the polynomial function $f(x) = x^3 - 6x^2 + 9x$ and study its sign across the real number line, stating the intervals where the function is positive and negative.

Determine the analytical expression of a degree 4 polynomial function $P(x)$ that simultaneously satisfies the following conditions:\n\n1. It is an even function.\n2. It has exactly three distinct real roots, one of them being $x = 2$.\n3. Its graph intersects the horizontal line $y = 12$ at the point with x-coordinate $x = 1$.\n\na) Deduce what the other two real roots of $P(x)$ must necessarily be, justifying your deduction from the symmetry of even functions and the definition of a root.\nb) Write the factored form of $P(x)$ and calculate the value of its leading coefficient using the condition of the given point.\nc) Expand the polynomial and describe its end behavior (as $x \to +\infty$ and as $x \to -\infty$), justifying it using the degree and the sign of the leading coefficient.

An intercity transport company operating the route between Madrid and Toledo currently charges $10\text{ euros}$ per ticket and records an average of $1200$ daily passengers. According to a market study, for each $0,50\text{ euro}$ decrease in the ticket price, the company attracts $100$ new passengers each day.\n\na) Model the daily revenue function $I(x)$ in terms of the number $x$ of $0,50\text{ euro}$ discounts applied. Determine the domain of definition of $x$ that makes physical and economic sense in this real-world context.\nb) Justify analytically, using the properties of the vertex of the resulting polynomial function, what the final ticket price should be to maximize daily revenue and what that revenue will amount to.\nc) If the company decided to set an aggressive promotional fare of only $2\text{ euros}$ per ticket, what revenue would the model predict? Explain why this quadratic model ceases to be realistic for such extreme price reductions.

Find the algebraic expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, given also that its graph intersects the y-axis at the point $(0, 16)$.

A technology company sells $x$ units of a certain electronic component. The unit selling price modelled in euros is $p(x) = 200 - 0,5x$. Total production costs follow the function $C(x) = 5000 + 20x$. Express the profit function $B(x)$, determine the number of units that maximises profit, and calculate this maximum profit.

Determine the values of the parameter $m$ for which the line with equation $y = mx - 4$ is tangent to the parabola $y = x^2 - 3x + 5$.

The main cable of a suspension bridge takes the shape of a parabola. The horizontal distance between the support towers is $120$ metres, and the lowest point of the cable is $20$ metres below the attachment point on the towers. Taking the lowest point of the cable as the origin of coordinates, find the equation of the parabola and calculate the height of the cable $30$ metres from the centre.

A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Given that it passes through the points $(1, -4)$ and $(2, 4)$, determine its algebraic expression and calculate all its axis intercepts.

A rectangular piece of cardboard measuring $40$ cm long by $30$ cm wide is available. An open-topped box is to be made by cutting out a square of side length $x$ from each corner and folding up the resulting flaps. Express the volume $V(x)$ of the box as a polynomial function, state its valid domain in the context of this geometric problem, and calculate the volume if a square of side length $5$ cm is cut out.

Calculate the set of values of the parameter $k$ for which the polynomial function $P(x) = x^4 - 2x^2 + k$ has exactly two distinct real roots.

Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin of coordinates and that its vertex lies on the line $y = -x - 2$, find the analytical expression of the function or functions that satisfy these conditions.

To study the intervals of increase and decrease and the range of the quadratic function $f(x) = 3(x - 5)^2 - 12$, three students propose different methods:\n\n- Lucía proposes expanding the expression into the general form $f(x) = ax^2 + bx + c$, calculating the x-coordinate of the vertex using the formula $x_v = -\dfrac{b}{2a}$, and then finding $f(x_v)$.\n- David argues that there is no need to expand anything: he observes that $(x - 5)^2 \ge 0$ for any real number and proposes reading the coordinates of the vertex directly from the given vertex form.\n- Álvaro proposes solving the equation $f(x) = 0$ to find the two x-intercepts and calculating their arithmetic mean to locate the axis of symmetry of the parabola.\n\na) Apply all three procedures to find the vertex and the range of $f(x)$.\nb) Compare the three methods in terms of computational economy and robustness. Justify which one is the most efficient in this specific case.\nc) Now consider the function $g(x) = 3(x - 5)^2 + 12$. Analyze which of the three methods would be invalidated if we work exclusively with real numbers, and explain why David's reasoning is conceptually superior for justifying the absolute minimum of the parabola.

In a review session on cubic polynomial functions of the form $f(x) = ax^3 + bx^2 + cx + d$ with $a \neq 0$, Daniel makes two categorical statements:\n\n- Statement 1: "Since any cubic polynomial with $a > 0$ tends to $-\infty$ to the left and to $+\infty$ to the right, its graph must always intersect the x-axis three times".\n- Statement 2: "If we eliminate the constant term ($d = 0$), the resulting function $f(x) = ax^3 + bx^2 + cx$ passes through the origin and, therefore, is always an odd function with symmetry with respect to the origin".\n\na) Refute Statement 1 by proposing an explicit counterexample of a cubic polynomial with $a > 0$ that intersects the x-axis at a single point, and justify via its increasing behavior why it cannot have more real roots.\nb) Prove analytically that Statement 2 is false by checking whether or not the identity $f(-x) = -f(x)$ holds for $f(x) = ax^3 + bx^2 + cx$. What necessary and sufficient condition must the coefficient $b$ satisfy for the function to truly be odd?\nc) Construct the algebraic expression of a cubic polynomial with leading coefficient $a = 1$ that is symmetric with respect to the point $(2, 5)$ and whose only real root is $x = 0$. Justify step by step how you translate the geometric conditions into the coefficients of the polynomial.