Exercises: Quadratic 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 $y = -x^2 + 2x + 8$.

Look at the three parabolas represented in the following figure. Match each of the analytical 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 way the parabola $y = -3x^2 + 6x - 1$ opens, 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 inches that a marble travels when rolling down an inclined track as a function of time $t$ in seconds. What distance will it have traveled after $4$ seconds? How long will it take to travel exactly $125$ in?

Express the function $y = (x + 3)^2 - 2$ as the result of applying a translation to the parent parabola $y = x^2$. Clearly state the direction and the number of units shifted on both the horizontal and vertical axes.

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

During a training session, a volleyball player throws the ball upwards. The height $h$ (in feet) 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?

A company that manufactures personalized cell phone cases has determined that if it sells each case for $\$x$, its daily revenue in dollars is given by the quadratic function $I(x) = -2x^2 + 48x$. At what price must they sell the cases to maximize their daily revenue? What does this maximum revenue amount to?

Find analytically 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 on 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 $40$ feet of rope to mark out a rectangular play area in a camp. If we call $x$ the length of the base of the rectangle, write the function that determines the area of the enclosure as a function of $x$. What must the dimensions of the rectangle be so that the enclosed area is as large as possible?

An arch of a stone bridge has the shape of a parabola. The base of the arch (at water level) is $12$ feet wide and the maximum height in the center is $6$ feet. If we place the origin of coordinates at the center 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 $(-1, -2)$.

A farmer wants to build a rectangular pen for his chickens using an existing long stone wall, so that he only needs to fence three of the four sides of the rectangle. He has $100$ feet of wire fencing. Determine the dimensions of the pen so that its area is as large as possible and calculate this maximum area.

Draw a sketch of the graph of the function $f(x) = |x^2 - 6x + 5|$. To do this, first find the vertex and the points of intersection with the axes of the inner parabola without the absolute value, and explain analytically how the absolute value affects the final representation.

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).

An electric scooter factory has fixed monthly expenses of $\$2000$ and producing each scooter costs an additional $\$15$. The monthly revenue obtained from the sale of $x$ scooters is given by the function $I(x) = -0.1x^2 + 65x$. Write the function that represents the monthly profit $B(x)$. How many scooters must they manufacture and sell to obtain the maximum profit?

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

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 breakpoint $x = 3$ and, in addition, the half-line passes through the point $(5, -1)$.