Introduction This diagram shows the locations of the three hospitals, $A$, $B$ and $C$ in a city. When a car accident occurs, it is important to locate the nearest hospital. How could we improve the diagram to make it easier to identify the closest hospital to any given location? If it is decided that a […]

# Graphs

# Stretches of Graphs

Stretch Rule 1 For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$. If $p \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis. If $0 \lt p \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis. Stretch Rule 2 […]

# Reflections of Graphs

For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis. For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis. Example 1 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$. \( \begin{align} \displaystyle f(x) &= x^3-4x^2+4x \\ &= x(x^2 – 4x + 4) \\ &= x(x-2)^2 \end{align} \) Example 2 Consider $f(x)=x^3-4x^2+4x$. On the […]

# Transformation of Rational Functions

Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote. Any graph of a rational function can be obtained from the reciprocal function $f(x)=\dfrac{1}{x}$ by a combination of transformations including a translation, stretches and compressions. $y=\dfrac{a}{x}$: horizontal compression, $0 \lt a \lt 1$ $y=\dfrac{a}{x}$: horizontal stretch, $a \gt 1$ […]

# Translations of Graphs

Translation Rule 1 For $y=f(x)+b$, the effect of $b$ is to translate the graph vertically through $b$ units. If $b \gt 0$, it moves upwards. If $b \lt 0$, it moves downwards. Translation Rule 2 For $y=f(x-a)$, the effect of $a$ is to translate the graph horizontally through $a$ units. If $a \gt 0$, it […]

# Reciprocal Functions

These techniques involves sketching the graph of $y=\dfrac{1}{f(x)}$ from the graph of $y=f(x)$. Technique 1 When $f(x)$ approaches towards $0$, $y=\dfrac{1}{f(x)}$ approaches towards $\infty$, the graph of $y=\dfrac{1}{f(x)}$ approaches the vertical asymptote(s). Technique 2 The graph of $y=\dfrac{1}{f(x)}$ has vertical asymptotes at the $x$-intercepts of $y=f(x)$. Technique 3 When $f(x)$ approaches towards $\infty$, $y=\dfrac{1}{f(x)}$ approaches […]

# Where Graphs Meet

Suppose we sketch the graphs of two functions $f(x)$ and $g(x)$ on the same axes. The $x$-coordinates of points where the graphs meet are the solutions to the equation $f(x)=g(x)$. We can use this property to solve equations graphically, but we must make sure the graphs are drawn carefully and accurately. Let’s take a look […]