2.4 Key features of graphsIB Maths: Analysis and Approaches HL: Revision notes

The $y$-intercept is where $x = 0$: the point $(0, f(0))$. The $x$-intercepts are where $f(x) = 0$.

The $x$-values where $f(x) = 0$ are the zeros of the function $f$; they are also the roots of the equation $f(x) = 0$. For $f(x) = \frac{2x - 6}{x + 1}$ the fraction is zero when the numerator is zero, so $x = 3$.

If a factor is repeated, e.g. $h(x) = 0.02x(x - 15)^2$, the graph touches the axis at $x = 15$ instead of crossing it.

A local maximum is a point higher than all nearby points; a local minimum is lower than all nearby points. A parabola has one turning point, its vertex.

Use a GDC's maximum/minimum tool to find them, and write coordinates to 3 s.f. The maximum value is the $y$-coordinate, not the $x$-coordinate.

On a restricted domain, compare turning points with the end points. For $h(x) = 0.02x^3 - 0.6x^2 + 4.5x$, $0 \le x \le 20$: local maximum $(5, 10)$, local minimum $(15, 0)$, and $h(20) = 10$, so the range is $0 \le h \le 10$.

To find the greatest gap between two curves, graph their difference and find its maximum.

Look for symmetry to save work and to check answers.

• If $f(-x) = f(x)$ (e.g. only even powers of $x$, like $x^4 - 8x^2 + 3$), the graph is symmetric in the $y$-axis: if $(2, -13)$ is on it, so is $(-2, -13)$.
• If $f(-x) = -f(x)$ (e.g. $x^3 - 4x$), the graph has rotational symmetry of order 2 about the origin: $(a, b)$ gives $(-a, -b)$.
• A parabola is symmetric about the vertical line through its vertex, its axis of symmetry.

An asymptote is a line the graph approaches more and more closely.

• Vertical asymptote $x = a$: the function is undefined at $a$ and $|f(x)|$ grows without limit near it. For a fraction this is where the denominator is zero (and the numerator is not): $\frac{2x - 6}{x + 1}$ has $x = -1$.
• Horizontal asymptote $y = b$: $f(x)$ approaches $b$ as $x \to \pm\infty$. For $\frac{ax + b}{cx + d}$ it is $y = \frac{a}{c}$, so $\frac{2x - 6}{x + 1}$ has $y = 2$.

With technology, look at a table of values for very large $x$, or zoom out, to see the horizontal asymptote. In a model, a horizontal asymptote is a long-run limit: typing speed $W(t) = \frac{60t + 20}{t + 2}$ approaches 60 words per minute but never reaches it.

To find where two graphs meet, graph both on a GDC and use the intersect tool; or graph $f(x) - g(x)$ and find its zeros.

• Check the whole domain: $W(t) = \frac{60t + 20}{t + 2}$ and $V(t) = 15 + 4t$ meet twice, at $t = 0.279$ and $t = 8.97$.
• Between intersections one graph stays above the other, which tells you which quantity is larger.
• Give coordinates to 3 s.f. and interpret them with units.