internet intelligent tutors
Stretch Rule 1 For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$. Stretch Rule 2 For $y=f(kx)$, $k \gt 0$, the effect of $k$ is to horizontally compress the graph by a factor of $k$. Example 1 Given that the point $(-2,1)$ lies on […]
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 $y=-f(x)$. \( \begin{align} \displaystylef(x) &= x^3-4x^2+4x \\&= x(x^2-4x + 4) \\&= x(x-2)^2\end{align} \) Example 2 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of […]
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. Translation Rule 2 For $y=f(x-a)$, the effect of $a$ is to translate the graph horizontally through $a$ units. Translation Rule 3 For $y=f(x-a)+b$, the graph is translated $a$ units horizontally and vertically $b$ units. Example 1 Given […]
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 \( 3 \) […]