 # Stretches of Graphs $y=pf(x)$ and $y=f(qx)$

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 $y=-f(x)$ and $y=f(-x)$

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 $y=-f(x)$. Show Solution \begin{align} \displaystyle f(x) &= x^3-4x^2+4x \\ &= x(x^2 - 4x + 4) \\ &= x(x-2)^2 \end{align} Example 2 Consider $f(x)=x^3-4x^2+4x$. [...] # 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 $y=f(x)+b$ and $y=f(x-a)$

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 [...]