Question

The graph of function f is shown on the coordinate plane. Graph the line representing function g, if g is defined as shown below.
G(x)=1/2f(x+2)

Answer 1

Answer:

$g(x)=x-1$

Step-by-step explanation:

Any problem of this kind should first be approached by solving the initial function.

This question is based on curve translation on x-axis. But, there can be questions asked on translation in both axes, rotation of curve about origin, or both combined.  

Initial function f(x) is a straight line from picture.

Common equation of straight line is

$y=mx+c$

m= slope of the line

c= y-intercept

Slope of the line:

$m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}$

Let those points on graph be A(3,0) and B(0,-6)

Therefore, slope of line AB, using formula is:

$m=\frac{0-(-6)}{3-0}$

m=2

And c=(-6)

And equation of line AB is:

$y=2x-6$

$f(x)=2x-6$

Now this line is translated backwards by 2 units, (x) becomes (x+2)

Let us consider a X, such that, $X=x+2$

$f(X)=2X-6$

$f(x+2)=2(x+2)-6$

$f(x+2)=2x+4-6$

$f(x+2)=2x-2$

Now, there is another function such that, $g(x)= \frac{1}{2} f(x+2)$

$g(x)= \frac{1}{2}(2x-2)$

$g(x)= x-1$ ; according to your question.

In the picture, they are asking for a function, $g(x)= \frac{-1}{2} f(x+2)$

Then, $g(x)= -x+1$ ; according to the picture.

Here, I have attached a file showing all the graphs for clear understanding.

Answer 2

Answer:

See attachment below.

Step-by-step explanation:

f is a first-order polynomial, whose form is:

$y = m\cdot x + b$

Where:

$x$ - Independent variable.

$y$ - Dependent variable.

$m$ - Slope.

$b$ - y-Intercept.

The y-Intercept is equal to the value of the dependent variable when the independent variable is zero. Then:

$b = -6$

The slope is the ratio of the change in the dependent variable to the change in the independent variable. Hence:

$m =\frac{\Delta y}{\Delta x}$

$m = \frac{0-(-3)}{3-0}$

$m = 1$

The equation for f is:

$f(x) = x -6$

The expression for g is found after substituting into and manipulating f:

$g(x) = -\frac{1}{2}\cdot [(x+2)-6]$

$g(x) = -\frac{1}{2}\cdot (x-4)$

$g(x) = -\frac{1}{2}\cdot x + 2$

The graph of the new function is plotted herein.