Exercises: Polynomial Functions

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 analytical expression of the linear function that passes through the points $A(2, -3)$ and $B(5, 6)$.

Calculate analytically the points of intersection with the coordinate axes of the polynomial function $p(x) = x^3 - 4x$.

The height in feet 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 the instants $t = 1$ and $t = 3$.

State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ opens downward or opens upward and calculate the exact coordinates of its vertex.

Analyze the end behavior of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$, 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 $100$ feet of fencing to enclose a rectangular vegetable patch using the straight bank of a river, so no fencing is needed on that side. Express the area of the patch in terms of the width $x$ perpendicular to the river and determine the dimensions that maximize this area.

A projectile is launched upwards so that its height in feet 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 instant of time at which it occurs.

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

The selling price in dollars of an item as a function of the number of units $x$ demanded in 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 maximize the revenue.

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

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

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

Find the analytical expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, knowing 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 dollars is $p(x) = 200 - 0.5x$. The total production costs follow the function $C(x) = 5000 + 20x$. Express the profit function $B(x)$, determine the number of units that maximizes the profit and calculate the amount of 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 $120$ feet and the lowest point of the cable is $20$ feet 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 at $30$ feet from the center.

A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Knowing that it passes through the points $(1, -4)$ and $(2, 4)$, determine its analytical expression and calculate all its points of intersection with the coordinate axes.

A rectangular sheet of cardboard $40$ in long by $30$ in wide is available. We want to build a box without a lid by cutting out a square of side $x$ from each of the corners 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 $5$ in is cut out.

Calculate the set of values of the parameter $k$ so that 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 and that its vertex lies on the line $y = -x + 1$, find the algebraic expression for the function or functions that satisfy these conditions.