IB Maths AI · Topic 2 · 32 cards

Functions flashcards

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What makes a relation a function, and how can you test it on a graph?
Each input gives exactly one output. On a graph, no vertical line crosses the curve more than once (vertical line test).
What are the domain and the range of a function?
The domain is the set of allowed inputs (x-values); the range is the set of resulting outputs (y-values). For \(f(x)=\sqrt{x}\): domain \(x\ge0\), range \(y\ge0\).
What does \(f(3)\) mean, and how is it different from solving \(f(x)=7\)?
\(f(3)\) means substitute \(x=3\) to get an output. Solving \(f(x)=7\) means finding the input(s) that give output 7. If \(f(x)=2x^2-1\), then \(f(3)=17\).
How do you find the gradient between two points? Find it for (1, 2) and (4, 11).
\(m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{11-2}{4-1}=3\).
What are the common forms of the equation of a straight line?
Gradient-intercept \(y=mx+c\); point-gradient \(y-y_1=m(x-x_1)\); general \(ax+by+d=0\).
How are the gradients of parallel and perpendicular lines related? What is the perpendicular gradient to \(\tfrac23\)?
Parallel: \(m_1=m_2\). Perpendicular: \(m_1m_2=-1\). The gradient perpendicular to \(\tfrac23\) is \(-\tfrac32\).
What does the gradient mean in a real-world linear model, and what does the intercept mean?
The gradient is the rate of change (units: y-units per x-unit). The y-intercept is the starting value when \(x=0\). Always state both with units in context.
What is the axis of symmetry of \(f(x)=ax^2+bx+c\), and what is the vertex of \(x^2-6x+5\)?
Axis: \(x=-\dfrac{b}{2a}\). For \(x^2-6x+5\) the axis is \(x=3\) and \(f(3)=-4\), so the vertex is \((3,-4)\).
What does each quadratic form reveal: \(a(x-p)(x-q)\) and \(a(x-h)^2+k\)?
Factored form shows the roots \(x=p,q\). Vertex form shows the vertex \((h,k)\). The standard form \(ax^2+bx+c\) shows the y-intercept \(c\).
How does the discriminant \(b^2-4ac\) tell you the number of roots? For what \(k\) does \(x^2+4x+k=0\) have equal roots?
\(\Delta>0\): two real roots; \(\Delta=0\): one repeated root; \(\Delta<0\): none. Equal roots need \(16-4k=0\), so \(k=4\).
What is an exponential model \(f(x)=ka^x+c\), and what do \(a\) and \(c\) control?
If \(a>1\) it shows growth, if \(0
How does a continuous growth model \(P=P_0e^{kt}\) work? Find \(P\) when \(P_0=500\), \(k=0.03\), \(t=10\).
\(e\approx2.718\), and \(k\) is the continuous growth rate per unit time. \(P=500e^{0.3}\approx675\).
What is a half-life model, and what is left of 80 g after 15 days if the half-life is 5 days?
\(A=A_0\left(\tfrac12\right)^{t/h}\). Here \(15\div5=3\) half-lives: \(80\times\tfrac18=10\) g.
What are direct and inverse variation? If \(y\) is inversely proportional to \(x\) and \(y=6\) at \(x=4\), find \(y\) at \(x=8\).
Direct: \(y=kx\) (straight line through the origin). Inverse: \(y=\dfrac kx\) (hyperbola). Here \(k=24\), so at \(x=8\), \(y=3\).
What are the features of a cubic function \(ax^3+bx^2+cx+d\)?
At most 3 real roots and at most 2 turning points; it always crosses the x-axis at least once; ends go opposite ways to \(\pm\infty\).
What do \(a\), \(b\), \(c\), \(d\) mean in \(f(x)=a\sin(b(x-c))+d\)? Give the amplitude, period and range of \(3\sin(2x)+1\).
Amplitude \(|a|\), period \(\dfrac{2\pi}{b}\) (or \(\dfrac{360^\circ}{b}\)), phase shift \(c\), midline \(y=d\). For \(3\sin2x+1\): amplitude 3, period \(\pi\), range \(-2\le y\le4\).
How do you find where two graphs meet using the GDC, and what does it solve?
Graph both functions and use the intersection tool. The x-coordinates solve \(f(x)=g(x)\). Setting \(y=0\) and finding x-intercepts gives the zeros of \(f\).
How do you find the horizontal and vertical asymptotes of \(f(x)=\dfrac{2x+1}{x-3}\)?
Vertical: where the denominator is 0, so \(x=3\). Horizontal: the value as \(x\to\pm\infty\), here the ratio of leading coefficients, \(y=2\).
What is the average rate of change of \(f(x)=x^2\) between \(x=1\) and \(x=3\)?
\(\dfrac{f(3)-f(1)}{3-1}=\dfrac{9-1}{2}=4\). It is the gradient of the chord joining the two points.
What are the stages of the modelling process?
Pick a model type from the shape of the data, fit it (GDC regression), interpret the parameters in context, test the fit against the data, then state limits such as domain and validity range.
Which models can a GDC fit, and how do you choose between them?
Linear, quadratic, cubic, exponential, power and sinusoidal. Match the shape of the scatter plot to the model, then compare \(R^2\) and residual patterns, and check the model makes sense outside the data.
Why should a model's domain be stated, and what is an example?
Real situations have limits: time cannot be negative and a population cannot exceed what can exist. For example, a model of height against age may only be valid for \(0\le t\le18\).
(HL) What is a composite function \((f\circ g)(x)\)? If \(f(x)=x+2\) and \(g(x)=3x\), find \(f(g(2))\) and \(g(f(2))\).
\((f\circ g)(x)=f(g(x))\): apply \(g\) first. \(f(g(2))=f(6)=8\), but \(g(f(2))=g(4)=12\), so order matters.
(HL) How do you find the inverse \(f^{-1}\), and when does it exist? Find it for \(f(x)=2x+3\).
Swap \(x\) and \(y\), then rearrange. It exists only if \(f\) is one-to-one (the domain may need restricting). \(f^{-1}(x)=\dfrac{x-3}{2}\).
(HL) How is the graph of \(f^{-1}\) related to the graph of \(f\), and how do their domains and ranges compare?
It is the reflection of \(f\) in the line \(y=x\). The domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\).
(HL) What does \(f(x-a)+b\) do to the graph of \(f(x)\)?
It translates the graph \(a\) units right and \(b\) units up, i.e. by the vector \(\begin{pmatrix}a\\b\end{pmatrix}\). Note the sign: \(x-a\) moves right.
(HL) What do \(af(x)\), \(f(bx)\), \(-f(x)\) and \(f(-x)\) do to a graph?
\(af(x)\): vertical stretch by factor \(a\). \(f(bx)\): horizontal stretch by factor \(\tfrac1b\). \(-f(x)\): reflection in the x-axis. \(f(-x)\): reflection in the y-axis.
(HL) What is a logistic model \(f(x)=\dfrac{L}{1+Ce^{-kx}}\), and what does \(L\) represent?
An S-shaped growth curve. \(L\) is the limiting (carrying-capacity) value, the upper horizontal asymptote. Growth is fastest when the value is \(\tfrac L2\).
(HL) What is a piecewise function, and what is \(f(5)\) if \(f(x)=2x\) for \(x<3\) and \(f(x)=x+3\) for \(x\ge3\)?
A function with different rules on different intervals. Pick the rule whose interval contains the input: \(5\ge3\), so \(f(5)=5+3=8\).
(HL) What are the key features of \(y=\ln x\)?
It is the inverse of \(e^x\). Domain \(x>0\), range all reals, vertical asymptote \(x=0\), passes through \((1,0)\), and increases slowly.
(HL) How do you solve \(e^{x}=20\)?
Take natural logs of both sides: \(x=\ln20\approx3.00\). In general, \(a^x=b\) gives \(x=\dfrac{\ln b}{\ln a}\).
(HL) How are \(e^k\) and a yearly percentage growth rate \(r\) linked in \(P_0e^{kt}\) and \(P_0(1+r)^t\)?
The models agree when \(e^{k}=1+r\), so \(k=\ln(1+r)\). For 5% yearly growth, \(k=\ln1.05\approx0.0488\).

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