Representing functions

A line passes through the points $(-3, -4)$ and $(0, 4)$. Determine its slope, its y-intercept, and its algebraic expression.

Find the domain of definition of the radical function $f(x) = \sqrt{2x - 8}$.

A taxi driver charges a standing charge of $3$ € and $1,5$ € for each kilometre travelled. Write the algebraic expression for the function relating the cost of the journey to the distance travelled in kilometres.

Determine the coordinates of the vertex and the equation of the axis of symmetry of the parabola $f(x) = x^2 - 6x + 5$.

Starting from the base parabola $y = x^2$ (dashed line), a translation has been applied to obtain the parabola shown as a solid line. Write down the algebraic expression of the new parabola and state the translation vector applied.

State the vertical and horizontal asymptotes of the inverse proportion function $f(x) = \frac{2}{x-3} + 4$.

In a quiz game, the points accumulated double after each completed round. If you start with $100$ points, write down the exponential function $P(r)$ that gives the points as a function of the number of completed rounds $r$.

Determine the domain of the logarithmic function $f(x) = \log_3(x - 5)$.

An amount of $5000$ € is invested in a bank deposit offering an annual compound interest rate of $4\%$. Write the exponential function that models the accumulated capital after $t$ years and calculate the balance available after $5$ years.

Given the piecewise function $f(x) = \begin{cases} 2x+1 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$, calculate the values of $f(-2)$, $f(0)$ and $f(3)$.

The following graph shows a cyclist's speed over time. Determine the algebraic expression of the piecewise function that represents it.

The height $h(t)$ in metres of a Ferris wheel cabin above the ground as a function of time $t$ in minutes is given by $h(t) = 12 + 10\sin\left(\frac{\pi}{4}t\right)$. What are the maximum and minimum heights reached by the Ferris wheel? How long does it take to complete one full revolution?

Express the function $f(x) = |x^2 - 5x + 6|$ as a piecewise function without using the modulus. State the intervals in which the original quadratic function was negative before applying the modulus.

The trajectory of a projectile follows a parabola with equation $y = ax^2 + bx + c$. Given that the projectile is launched from point $(0, 3)$, reaches its maximum height at point $(2, 7)$, and hits the ground, find the algebraic expression of the parabola and determine the horizontal distance from the launch point at which it lands.

The graph shows the intersection between the line $y = -x + 5$ and the hyperbola $y = \frac{4}{x}$. Calculate algebraically the exact coordinates of the points of intersection $A$ and $B$.

Determine the values of the parameters $a$, $b$ and $c$ of the rational function $f(x) = \frac{ax + 2}{bx + c}$, given that it has a vertical asymptote at $x = 1$, a horizontal asymptote at $y = 3$, and passes through the point $(2, 8)$.