Question

The graph of f(x)=cos(x) is transformed to a new function, g(x) , by stretching it horizontally by a factor of 4 and shifting it 1 unit up. What is the equation of the new function g(x) ?

Answer 1

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ % function transformations for trigonometric functions \begin{array}{rllll} % left side templates f(x)=&{{ A}}cos({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ \end{array}$

$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks}\\ \quad \textit{horizontally by amplitude } |{{ A}}|\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\ \end{array}$

$\bf \begin{array}{llll} \bullet \textit{vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\ \bullet \textit{function period}\\ \qquad \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\ \qquad \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta) \end{array}$

stretched by factor of 4? A=1/4
vertical shift of 1? D = 1

Answer 2

Answer:

The equation formed is $g(x)=\cos(\frac{x}{4})+1$.

Step-by-step explanation:

Given : The graph of f(x)=cos(x) is transformed to a new function, g(x) , by stretching it horizontally by a factor of 4 and shifting it 1 unit up.

To find : What is the equation of the new function g(x) ?

In horizontally stretch, $(x,y)\rightarrow (\frac{x}{b},y)$

Stretching it horizontally by a factor of 4.

i.e. $g(x)=\cos(\frac{x}{4})$

Shifting 'a'unit up f(x)→f(x)+a

i.e. $g(x)=\cos(\frac{x}{4})+1$

So, the equation formed is $g(x)=\cos(\frac{x}{4})+1$.