# Tag Archives: Sketching 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

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 […] # Translations of Graphs

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 […] # 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 […] Consider the graphs of a quadratic function and a linear function on the same set of axes. cutting2 points of intersection $b^2-4ac \gt 0$ touching1 point of intersection $b^2-4ac = 0$ missingno points of intersection $b^2-4ac \lt 0$ In the graphs meet, the coordinates of the points of intersection of the graphs can be found […] 