2.4 Key features of graphsIB Maths: Applications and Interpretation SL: Revision notes

A maximum or minimum value is the highest or lowest value of the function in its domain. A turning point is where the graph changes from increasing to decreasing or the other way round. For a quadratic the turning point is the vertex. Use your GDC's maximum or minimum tool and give both coordinates: the $x$-value says when or where, and the $y$-value says how big. Example: $A(x)=x(60-2x)$ has its vertex at $(15, 450)$, so the largest area is 450 m$^2$ when the width is 15 m. On a restricted domain the greatest or least value may be at an end point, so compare the turning points with the values at the ends.

The $y$-intercept is the point where $x=0$. The zeros of $f$ are the $x$-values where $f(x)=0$; they are the $x$-intercepts of the graph, and the roots of the equation $f(x)=0$. Use your GDC's zero (root) tool, or the equation solver. Example: $A(x)=x(60-2x)$ has zeros $x=0$ and $x=30$. In context, $x=30$ would leave no fencing for the length, so the area is zero.

The graph of a quadratic $y=ax^2+bx+c$ is symmetric about a vertical line through its vertex, the axis of symmetry, $x=-\frac{b}{2a}$. It lies halfway between the two zeros. Example: if the zeros are $x=0$ and $x=30$, the axis of symmetry is $x=15$. For $A(x)=-2x^2+60x$: $-\frac{60}{2(-2)}=15$. Symmetry lets you find the vertex from the zeros: the $x$-value is the midpoint.

An asymptote is a line that the graph gets closer and closer to but does not reach. A vertical asymptote is where the function is undefined, e.g. $n=0$ for $C(n)=\frac{600}{n}+15$. A horizontal asymptote shows the long-term value as $x$ becomes large. For $C(n)=\frac{600}{n}+15$: as $n\to\infty$, $\frac{600}{n}\to0$, so the horizontal asymptote is $C=15$. For an exponential model such as $T=20+70\times0.9^t$, the asymptote is $T=20$. View the graph on your GDC and write the asymptote as an equation.

To find where two curves or lines meet, graph both on your GDC and use the intersection tool. The $x$-coordinates satisfy $f(x)=g(x)$. Example: $P(t)=500\times1.3^t$ and $Q(t)=2000+300t$ meet at $t=8.39$. You can also graph the difference, $Q(t)-P(t)$, and find its zeros and its maximum: the greatest gap is $1800$ at $t=3.15$. Give answers to 3 significant figures and interpret them in context, with units.