Exercises: Graphing functions

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 $f(x) = x^2 - 6x + 5$.

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 $r$.

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 $f(3)$.

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

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

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

Marcos has to plot the rational function $f(x) = \dfrac{3}{x-2} + 4$. In his notebook he writes: “Since the denominator vanishes at $x = 2$, the horizontal asymptote is $x = 2$. And since the constant added at the end is $4$, the vertical asymptote is $y = 4$. Therefore, the graph cannot intersect either of the two coordinate axes.” Identify the two conceptual mistakes Marcos made, explain in your own words why he confuses the behaviour of the asymptotes, and correct his analysis by determining the correct equations of both asymptotes and the coordinates of the $Y$-intercept.

Find the algebraic expression of a quadratic function of the form $f(x) = ax^2 + bx + c$ that simultaneously satisfies these three geometric conditions: its vertex is at the point $V(3, 4)$, it intersects the $y$-axis (the axis of ordinates) at the point $(0, -5)$, and its branches open downwards. Explain step by step how you deduce the coefficients $a$, $b$, and $c$ from the symmetry of the parabola and the properties of the vertex, without merely mechanically solving a system of three equations with three unknowns.

The profit from selling end-of-year party T-shirts, in euros, is given by the function $B(x) = -2x^2 + 120x - 1000$, where $x$ is the number of T-shirts sold in hundreds. How many T-shirts must be sold to maximise profit, and what is this maximum profit?

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

Hiring a coach for a school trip costs $600$ €. Write down the function $C(x)$ that gives the cost per student depending on the number of students $x$ attending. If $5$ students cancel at the last minute, write the new cost-per-student function and find how much each student will pay if $30$ students were originally going.

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.

The time $t$ (in seconds) taken for a dropped object to fall to the ground from a height $h$ (in metres) is approximated by the radical function $t(h) = \sqrt{\frac{2h}{9,8}}$. Find the domain of the function in this context and determine the time taken for an object dropped from a height of $4,9$ metres to fall.

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.

The sound intensity level $L$, in decibels (dB), is calculated using the logarithmic function $L(I) = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity and $I_0 = 10^{-12} \text{ W/m}^2$ is the threshold of hearing. Calculate the sound intensity level of a normal conversation with an intensity of $I = 10^{-6} \text{ W/m}^2$.

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?

In a school athletics event in Spain, an athlete's shot put follows a parabolic trajectory given by the function $h(x) = -0,05x^2 + 0,6x + 1,8$, where $x$ is the horizontal distance travelled in metres and $h(x)$ is the height of the shot above the ground in metres.

1. a) Before calculating with formulas, make a reasoned estimate of between which integer values the distance at which the maximum height is reached will lie, and whether the final mark will exceed $14\text{ m}$, relying on the meaning of the $y$-intercept and the rate at which the trajectory changes.
2. b) Calculate with precision the coordinates of the vertex and the total range of the throw (intersection with the ground, $h(x) = 0$). Compare the numerical values obtained with your previous estimates and assess whether the mathematical model is consistent with the reality of this sport.

A public water supply company in Spain bills two-monthly domestic consumption to encourage responsible savings through a tiered rate. The standing charge for the service is $12$ € per two-month period. Water consumption is billed according to the following rules:

• The first $20\text{ m}^3$ consumed have a reduced price of $0,50$ € per $\text{m}^3$.

• For consumption greater than $20\text{ m}^3$ and up to a maximum of $50\text{ m}^3$, the first $20\text{ m}^3$ are billed at the reduced rate and each additional cubic metre exceeding $20$ costs $1,20$ €.

• Any consumption exceeding $50\text{ m}^3$ is charged by paying for the previous tiers plus $2,00$ € for each cubic metre that exceeds $50\text{ m}^3$.

Write the analytical expression of the piecewise function $C(x)$ that relates the total cost of the bill (in euros) as a function of the cubic metres $x$ consumed. Explain how you calculate the constant terms of each piece to ensure that the function is continuous and that no unjustified jump occurs in the amount.

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 $B$.

The amount of Carbon-14 in a fossil remnant decays exponentially according to the function $N(t) = N_0 \cdot (0,5)^{\frac{t}{5730}}$, where $N_0$ is the initial amount and $t$ is the time in years. If a bone retains only $12,5\%$ of its original Carbon-14, how many years have passed?

A farmer has $120$ metres of fencing to enclose a rectangular plot of land adjacent to a straight river. Given that no fencing is needed along the riverbank, write down the function that gives the area of the plot as a function of the width $x$ perpendicular to the river. What dimensions maximise the area, and what is the maximum area?

Given the inverse proportion function $f(x) = \frac{3x - 1}{x + 2}$, find the expression for its inverse function $f^{-1}(x)$. Verify algebraically that the vertical and horizontal asymptotes are swapped between $f(x)$ and $f^{-1}(x)$.

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 $(2, 8)$.

The monthly water bill in a municipality is calculated using a piecewise function: a standing charge of $10$ € including the first $15 \text{ m}^3$, a cost of $0,5$ € per additional $\text{m}^3$ for consumption between $15 \text{ m}^3$ and $30 \text{ m}^3$, and a cost of $1,2$ € for each $\text{m}^3$ exceeding $30 \text{ m}^3$. Write the algebraic expression for the cost function $C(x)$ in terms of the total cubic metres consumed $x$.

To accurately plot the quadratic function $f(x) = 2x^2 - 12x + 10$ and determine its axis of symmetry and its vertex, two students defend different strategies:

• Irene proposes transforming the algebraic expression by completing the square to obtain the vertex form $f(x) = a(x - p)^2 + q$.

• Sergio proposes first solving the equation $2x^2 - 12x + 10 = 0$ using the quadratic formula, averaging the two roots to place the axis of symmetry at the midpoint, and substituting that $x$-coordinate into the function to find the vertex.

1. a) Apply both methods step by step to find the coordinates of the vertex and the axis of symmetry.
2. b) Compare the two approaches: what advantages and disadvantages does each have? Justify in which specific situation (for example, considering the sign of the discriminant of the equation) Sergio's method ceases to be directly applicable in the set of real numbers, whereas Irene's procedure continues to work without any hindrance.

An oceanographic buoy installed off the coast of Galicia monitors tidal fluctuations. The sea level height $h(t)$ (in metres) relative to a reference seabed follows a periodic behaviour with period $T = 12\text{ h}$. During a basic cycle $t \in [0, 12]$, the height is modelled by two parabolic sections:

• Low-tide section (around the minimum level), for $t \in [0, 6]$: $h(t) = a(t - 3)^2 + 1$.

• High-tide section (around the maximum level), for $t \in [6, 12]$: $h(t) = -a(t - 9)^2 + k$.

It is known from harbour records that the sea level varies continuously (without sudden jumps), that the minimum low tide is $1\text{ m}$, and that the maximum high tide is $5\text{ m}$.

1. a) Deduce the exact values of the parameters $a$ and $k$. Justify your reasoning using the coordinates of the vertices and the continuity condition at the time the piece changes ($t = 6\text{ h}$).
2. b) Without needing to define new formulas for subsequent times, determine with reasoning the sea level height at the times $t = 27\text{ h}$ and $t = 66\text{ h}$ based exclusively on the formal property of periodicity $h(t + T) = h(t)$.