Graphing functions
The same information can be expressed as a formula, as a table of values or as a graph: they are three representations of the same function, and being able to move from one to another is just as important as calculating with the formula. This section reviews how to construct and interpret those three representations for the families of functions that you will use constantly from now on.
From the table to the graph
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…
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.
Straight lines and parabolas
Cubic functions
A cubic $f(x)=ax^3+bx^2+cx+d$ crosses the y-axis at $d$ and can cross the x-axis up to three times. If $a>0$ the graph rises from left to right; if $a<0$ it falls. It has up to two turning points.
Translations of parabolas
The graph of $y=(x-h)^2+k$ is the graph of $y=x^2$ moved $h$ units right and $k$ units up: translation vector $(h, k)$. Its vertex is $(h, k)$.
Vertex using the roots
For $f(x)=ax^2+bx+c$, the vertex lies midway between the x-intercepts, so $x_v$ is the mean of the two roots. Then find $f(x_v)$.
Determine the coordinates of the vertex and the equation of the axis of symmetry of the parabola $f(x) = x^2 - 6x + 5$.
Determine the coordinates of the vertex and the equation of the axis of symmetry of the parabola $f(x) = x^2 - 6x + 5$.
A line passes through the points $(-3, -4)$ and $(0, 4)$. Determine its slope, its y-intercept, and its algebraic expression.
A line passes through the points $(-3, -4)$ and $(0, 4)$. Determine its slope, its y-intercept, and its algebraic expression.
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…
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.
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: Aoife…
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:
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Aoife proposes transforming the algebraic expression by completing the square to obtain the vertex form $f(x) = a(x - p)^2 + q$.
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Seán 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.
- a) Apply both methods step by step to find the coordinates of the vertex and the axis of symmetry.
- 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) Seán's method ceases to be directly applicable in the set of real numbers, whereas Aoife's procedure continues to work without any hindrance.
Exponential functions and other families
Asymptotes of $f(x)=\dfrac{k}{x-p}+q$
The denominator is zero at $x=p$, so the graph never reaches $x=p$: this is the vertical asymptote. For very large $x$ the fraction $\dfrac{k}{x-p}$ gets close to $0$, so $f(x)$ gets close to $q$: this is the horizontal asymptote $y=q$.
Domain of a logarithmic function
The function $\log_a(\text{something})$ only exists when the something is positive, because no power of a positive base gives zero or a negative number.
State the vertical and horizontal asymptotes of the inverse proportion function $f(x) = \frac{2}{x-3} + 4$.
State the vertical and horizontal asymptotes of the inverse proportion function $f(x) = \frac{2}{x-3} + 4$.
Determine the domain of the logarithmic function $f(x) = \log_3(x - 5)$.
Determine the domain of the logarithmic function $f(x) = \log_3(x - 5)$.
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…
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 $r$.
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…
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.
Piecewise and periodic functions
Domain of a square root function
The expression under a square root must be $\ge 0$, otherwise the root is not a real number.
Find the domain of definition of the radical function $f(x) = \sqrt{2x - 8}$.
Find the domain of definition of the radical function $f(x) = \sqrt{2x - 8}$.
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…
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.
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 $f(3)$.
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 following graph shows a cyclist's speed over time. Determine the algebraic expression of the piecewise function that represents it.
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…
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?
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…
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.
From the graph to the formula
Intersection of a line and a curve
To find where two graphs meet, set the two expressions equal and solve; then substitute back to find $y$. For a hyperbola $y=\frac{k}{x}$, multiply through by $x$ to get a quadratic.
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…
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…
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 $B$.