Tag Archives: Sketching Graphs

Reflections of Graphs

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 $y=-f(x)$. \( \begin{align} \displaystylef(x) &= x^3-4x^2+4x \\&= x(x^2-4x + 4) \\&= x(x-2)^2\end{align} \) Example 2 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of […]

Must-Know 10 Basic Translations of Rational Functions Explained

Must-Know 10 Basic Translations of Rational Functions Explained

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 […]

Reciprocal Functions

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 […]

Axis of Symmetry of Quadratic Graphs

Axis of Symmetry of Quadratic Graphs

The equation of the axis of symmetry of $y=ax^2+bx+c$ is $x=-\dfrac{b}{2a}$. Example 1 Find the equation of the axis of symmetry of $y=x^2+4x-2$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= -\dfrac{4}{2 \times 1} \\\therefore x &= -2\end{align} \) Example 2 Find the equation of the axis of symmetry of $y=-2x^2-12x+3$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= […]

X-intercepts Y-intercepts

X-intercepts Y-intercepts

$x$-intercepts when $y=0$$y$-intercepts when $x=0$ Example 1 Sketch the graphs of $y=x^2+x-2$ by stating $x$- and $y$-intercepts. $x$-intercepts when $y=0$\( \begin{align} \displaystylex^2+x-2 &= 0 \\(x+2)(x-1) &= 0 \\x &= -2, 1 \\\end{align} \)$y$-intercepts when $x=0$\( \begin{align} \displaystyley &= 0^2+0-2 \\&= -2 \end{align} \) Example 2 Sketch the graphs of $y=-x^2+3$ by stating $x$- and $y$-intercepts. […]