Graphing Logarithmic Functions

The inverse function of $y=a^x$ is $y=\log_{a}{x}$. Therefore $y=\log_{a}{x}$ is an inverse function, it is a reflection of $y=a^x$ in the line $y=x$. The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $0 \lt a \lt 1$: The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $a \gt 1$: \begin{array}{|c|c|c|} \require{color} \hline & y=a^x & \color{red}y =\log_{a}{x} \\ \hline [...]

Natural Exponential Graphs

$\textit{Natural Exponential Graphs}$ $$y=e^x$$ Natural Exponential Graphs also follow the rule of translations and transformations. Example 1 Sketch the graphs of $y=e^x$ and $y=-e^x$. Show Solution Reflected to the $x$-axis. Example 2 Sketch the graphs of $y=e^x$ and $y=-e^{-x}$. Show Solution Example 3 Sketch the graphs of $y=e^x$ and $y=e^{-x}$. Show Solution Example 4 Sketch [...]

Exponential Graphs

Functions of the form exponential, where the base is a positive real number other than 1 are called exponential graphs or exponential functions.

Quartic Graph Sketching is based on their concavities and x-intercepts to determine the basic shape of the quartic graphs. Question 1 of Quartic Graph Sketching Sketch the graph of $y=(x-1)(x+1)(x+2)(x-3)$. Show Solution Question 2 Sketch the graph of $y=x(x^2-x)(x-2)(x+3)$. Show Solution $=x(x-1)(x-2)(x+3)$ Question 3 Sketch the graph of $[...] Cubic Graph Sketching Basics of Cubic Graph Sketching Cubic Graph Sketching start considering from the \(x$-intercepts and whether the leading coefficient is either positive or negative. Question 1 Sketch the graph of $y=(x-1)(x+2)(x-3)$. Show Solution Question 2 Sketch the graph of $y=(x^2-2x)(x+3)$. Show Solution $y=(x^2-2x)(x+3) = x(x-2)(x+3)$ Question 3 Sketch the graph [...]

Sketching Quadratic Graphs using Transfomration 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 Show Solution (b)   $y=f(x)-3$; Transforming dowanwards by [...]