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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 […]
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 […]
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. Type 1: Vertical Compression \( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 \) The […]
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 […]
Sketching quadratic graphs are drawn based on \( y=x^2 \) graph for transforming and translating. Question 1 \(f(x) = (x-3)^2 \) is drawn and sketch the following graphs by transforming. (a) \( y = f(x)+2 \); Transforming upwards by \( 2 \) units (b) \( y=f(x)-3 \); Transforming dowanwards by \( 3 \) […]