Question

F(x) = x2 – 3x – 2 is shifted 4 units left. The result is g(x). What is g(x)?

Answer 1

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\\\ \begin{array}{rllll} % left side templates f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}(\mathbb{R})^{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}} sin\left({{ B }}x+{{ C}} \right)+{{ D}} \end{array}$

$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative} \\\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \end{array}$

$\bf \begin{array}{llll} \bullet \textit{ vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}} \end{array}$

now... your expression is not in vertex form, that's ok... to do a horizontal shift to the left by 4 units, we can simply, add the C  and B component to the "x" variable, C=4, B =1, that way the horizontal shift of C/B or 4/1 is just +4, giving us a horizontal shift to the left

$\bf f(x)=x^2-3x-2\impliedby \textit{let's change that for }f(1x+4) \\\\\\ f(1x+4)=(1x+4)^2-3(1x+4)-2 \\\\\\ f(x+4)=x^2+8x+16-3x-12-2 \\\\\\ f(x+4)=x^2-5x+2\impliedby g(x)$

Answer 2

Answer:

The required function is $g(x)=x^2+5x+2$.

Step-by-step explanation:

The given function is

$f(x)=x^2-3x-2$

The transformation of a function is defied as

$g(x)=f(x+a)+b$

Where a is horizontal shift and b is the vertical shift.

If a>0, then the graph of f(x) shifts a units left and if a<0, then the graph of f(x) shifts a units right.

Since the graph of f(x) shifts 4 units left, therefore a=4.

$g(x)=f(x+4)$

$g(x)=(x+4)^2-3(x+4)-2$

$g(x)=x^2+8x+16-3x-12-2$

$g(x)=x^2+5x+2$

Therefore the  required function is $g(x)=x^2+5x+2$.