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 meters 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 behavior 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 $(-1, 10)$.

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 $100$ meters 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 maximize this area.

A projectile is launched upward such that its height in meters 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 pesos of an item as a function of the number of units $x$ demanded on the market is given by the function $p(x) = 2\,400 - 40x$. Write the total revenue function $R(x)$ and calculate how many units must be sold to maximize 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.

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 modeled in pesos is $p(x) = 4\,000 - 10x$. Total production costs follow the function $C(x) = 100\,000 + 400x$. Express the profit function $B(x)$, determine the number of units that maximizes 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 $120$ meters, and the lowest point of the cable is $20$ meters 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 $30$ meters from the center.

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.