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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 […]
$$y=e^x$$ Natural Exponential Graphs also follow the rule of translations and transformations. Example 1 Sketch the graphs of $y=e^x$ and $y=-e^x$. Reflected to the $x$-axis. Example 2 Sketch the graphs of $y=e^x$ and $y=-e^{-x}$. Example 3 Sketch the graphs of $y=e^x$ and $y=e^{-x}$. Example 4 Sketch the graphs of $y=e^x$ and $y=e^{x+1}$. Translated to left. […]
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 \( y=(x-1)(x+1)(x+2)(x-3) \). Question 2 Sketch the graph of \( y=x(x^2-x)(x-2)(x+3) \). \( =x(x-1)(x-2)(x+3) \) Question 3 Sketch the graph of \( y=x^2(x-3)(x+3) \). Question 4 […]
Cubic Graph Sketching start considering the \(x\)-intercepts and whether the leading coefficient is either positive or negative. Question 1 Sketch the graph of \( y=(x-1)(x+2)(x-3) \). The graph cuts at \( x=1, x=-2, x=3 \). Question 2 Sketch the graph of \( y=(x^2-2x)(x+3) \). \( y=(x^2-2x)(x+3) = x(x-2)(x+3) \) The graph cuts at \( x=0, […]
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 \( 3 \) […]