Sequences, functions and graphs flashcards

Test yourself one card at a time: say the answer, flip, then mark it. Cards you know drop out; the rest come back until you know them. Your progress stays in this browser.

How do you find the nth term of a linear sequence such as 5, 8, 11, 14?
The common difference (3) is the coefficient of \(n\). Compare \(3n\) with the terms: 3, 6, 9, 12, so add 2. The nth term is \(3n+2\).
Is 101 a term of the sequence with nth term \(3n+2\)?
Solve \(3n+2=101\): \(n=33\), a whole number, so yes. If \(n\) were not a whole number, the answer would be no.
What is a geometric sequence? Give the nth term of 3, 6, 12, 24, ...
Each term is the previous one multiplied by a constant ratio \(r\) (here 2). The nth term is \(3\times2^{n-1}\).
(Higher) Find the nth term of the quadratic sequence 3, 9, 19, 33, ...
First differences 6, 10, 14; second differences 4, so the \(n^2\) coefficient is \(4\div2=2\). Subtract \(2n^2\) (2, 8, 18, 32) to leave 1 each time: \(2n^2+1\).
If \(f(x)=3x-2\), what does \(f(4)\) mean and what is it?
Replace \(x\) by 4: \(f(4)=3\times4-2=10\). \(f\) is a rule that turns each input into one output.
What are the domain and range of a function?
The domain is the set of allowed inputs \(x\); the range is the set of outputs. For \(f(x)=\dfrac1x\) the domain excludes \(x=0\).
(Higher) If \(f(x)=2x+1\) and \(g(x)=x^2\), find \(fg(x)\) and \(gf(x)\).
\(fg(x)\) means do \(g\) first: \(2x^2+1\). \(gf(x)\) means do \(f\) first: \((2x+1)^2\). They are usually different.
(Higher) Find the inverse of \(f(x)=\dfrac{x+4}{3}\).
Write \(y=\dfrac{x+4}{3}\), swap \(x\) and \(y\): \(x=\dfrac{y+4}{3}\). Rearrange for \(y\): \(f^{-1}(x)=3x-4\).
How do you calculate the gradient of a straight line? Find it through (1, 3) and (5, 11).
\(m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{11-3}{5-1}=2\). It is the change up divided by the change across.
What do \(m\) and \(c\) represent in \(y=mx+c\)?
\(m\) is the gradient and \(c\) is the \(y\)-intercept (where the line crosses the \(y\)-axis).
Find the equation of the line through (2, 5) with gradient 3.
\(y-5=3(x-2)\), so \(y=3x-1\). Check by substituting \((2,5)\).
Find the gradient of the line \(3x+2y=6\).
Rearrange to \(y=mx+c\): \(2y=-3x+6\), \(y=-\tfrac32x+3\). The gradient is \(-\tfrac32\).
How do you draw the line 2x + 3y = 12 quickly?
Find the intercepts. If \(x=0\), \(y=4\). If \(y=0\), \(x=6\). Plot \((0,4)\) and \((6,0)\), join them. The gradient is \(-\tfrac23\).
(Higher) How are the gradients of parallel and perpendicular lines related?
Parallel lines have equal gradients. For perpendicular lines \(m_1m_2=-1\), so a line perpendicular to \(y=2x+1\) has gradient \(-\tfrac12\).
Find the midpoint and length of the line joining A(1, 2) and B(7, 10).
Midpoint: \(\left(\tfrac{1+7}2,\tfrac{2+10}2\right)=(4,6)\). Length: \(\sqrt{6^2+8^2}=\sqrt{100}=10\).
How do you find the turning point of \(y=x^2-4x+3\) and what shape is the graph?
The \(x\)-coordinate is \(-\dfrac b{2a}=2\); then \(y=4-8+3=-1\). Turning point \((2,-1)\). Since \(a>0\) the graph is a \(\cup\) shape.
Which graph shapes go with y = x³, y = 1/x and y = 2ˣ?
\(y=x^3\): S-shaped cubic through the origin. \(y=\dfrac1x\): two curves in opposite quadrants that never touch the axes (asymptotes). \(y=2^x\): rising curve through \((0,1)\) approaching the \(x\)-axis on the left.
Given the graph of \(y=x^2-2x-3\), how do you solve \(x^2-2x-3=1\)?
Draw the line \(y=1\) on the same axes and read the \(x\)-values where it crosses the curve. Solving \(=0\) uses the \(x\)-axis crossings instead.
(Higher) Describe the effect on the graph of \(y=f(x)\) of \(y=f(x+3)\) and \(y=f(x)+3\).
\(y=f(x+3)\): translation 3 units left, \(\begin{pmatrix}-3\\0\end{pmatrix}\). \(y=f(x)+3\): translation 3 units up, \(\begin{pmatrix}0\\3\end{pmatrix}\). Changes inside the bracket act on \(x\) and are 'backwards'.
(Higher) What transformations are \(y=-f(x)\), \(y=f(-x)\), \(y=2f(x)\) and \(y=f(2x)\)?
\(-f(x)\): reflection in the \(x\)-axis. \(f(-x)\): reflection in the \(y\)-axis. \(2f(x)\): vertical stretch, scale factor 2. \(f(2x)\): horizontal stretch, scale factor \(\tfrac12\).
What does the gradient of a distance-time graph show? A line goes from (10 s, 20 m) to (30 s, 80 m).
The gradient is the speed; a horizontal line means stationary. Here speed \(=\dfrac{60}{20}=3\text{ m/s}\).
What do the gradient and area represent on a velocity-time graph?
Gradient is acceleration; area under the graph is distance travelled. A car accelerates from 0 to 12 m/s in 4 s then holds 12 m/s for 6 s: acceleration \(3\text{ m/s}^2\), distance \(24+72=96\text{ m}\).
(Higher) How do you estimate the gradient of a curve at a point?
Draw a tangent touching the curve at that point, then find the gradient of the tangent using a large triangle. This gives the instantaneous rate of change.
(Higher) Differentiate \(y=4x^3-5x+2\).
For \(y=ax^n\), \(\dfrac{dy}{dx}=nax^{n-1}\), and a constant disappears. So \(\dfrac{dy}{dx}=12x^2-5\).
(Higher) Find the gradient of \(y=x^2-3x\) at \(x=4\).
\(\dfrac{dy}{dx}=2x-3\). At \(x=4\) the gradient is \(8-3=5\).
(Higher) Find and classify the stationary points of \(y=x^3-3x\).
\(\dfrac{dy}{dx}=3x^2-3=0\) gives \(x=\pm1\). Second derivative \(6x\): positive at \(x=1\) so a minimum at \((1,-2)\); negative at \(x=-1\) so a maximum at \((-1,2)\).
(Higher) What is the equation of a circle with centre the origin and radius 5?
\(x^2+y^2=25\). In general \(x^2+y^2=r^2\). A tangent at a point is perpendicular to the radius to that point.
How do you show the region \(y>2x+1\) on a graph?
Draw \(y=2x+1\) as a dashed line (strict inequality; solid for \(\le\) or \(\ge\)). Test a point such as \((0,0)\): \(0>1\) is false, so shade the side that does not contain the origin.
For y = 3 × 2ˣ, find y when x = 4.
\(y=3\times2^4=3\times16=48\). Exponential graphs of the form \(y=ka^x\) pass through \((0,k)\).

Practise this topicNotes Questions (316)Test builder