Question

Please please I really need help on both of these Thank you

Answer 1

Answer:

f(x) has a vertical asymptote at x = 5 and a horizontal asymptote at y = 2.

$g(x) = \frac{1}{x-5}+2$ is shifted 5 units to the right and 2 units up from f(x), as shown in figure a.

Step-by-step explanation:

Question # 4 Solution

$f(x) = \frac{1}{x-5}+2$

Determining Vertical Asymptote:

Setting the denominator equal to zero, we can determine the value of vertical asymptote.

As,

    x - 5 = 0

    x = 5

So, x = 5 is the Vertical Asymptote:

Determining Horizontal Asymptote

$f(x) = \frac{1}{x-5}+2$

$f(x) = \frac{2x-10}{x-5}$

If the degree of both numerator and denominator is same, then the horizontal asymptote can be obtained by determining the ratio of leading coefficient of the nominator to the leading coefficient of denominator.

As the leading coefficient of nominator is 2, and the leading coefficient of denominator is 1. So, the ratio of the leading coefficient of nominator to the leading coefficient of denominator is will get the horizontal asymptote.

Hence, y = 2/1 → y = 2 will be the horizontal asymptote.

So, f(x) has a vertical asymptote at x = 5 and a horizontal asymptote at y = 2.

Question # 5 Solution

A translation makes any graph of function move up or down - Vertical Translation -  and right or left - Horizontal Translation.

For example, if we replace the graph of y = f(x) with the graph of y = f(x - 5), meaning the graph y = f(x) will shift right by 5 units  so it is f(x - 5).

Important Note: Adding moves the graph to the left; subtracting moves the graph to the right.

As $g(x) = \frac{1}{x-5}+2$ compare to the parent function f(x) = 1/x.

The graph of f(x) = 1/x is shown in figure a. If we compare the graph of f(x) = 1/x with the graph of g(x) = 1 ÷ (x - 5), meaning the graph g(x) = 1 ÷ (x - 5) is shift right by 5 units, as shown in figure (a).

Now, take $g(x) = \frac{1}{x-5}+2$, we observe that graph $g(x) = \frac{1}{x-5}+2$ is now 2 units up from f(x). Adding 2 units to the output shifts the graph up by 2 units, as shown in figure a.

Hence, $g(x) = \frac{1}{x-5}+2$ is shifted 5 units to the right and 2 units up from f(x). The figure a shows the complete result of the graph $g(x) = \frac{1}{x-5}+2$.

Keywords: graph shift, vertical asymptote, horizontal asymptote

Learn more about horizontal and vertical asymptotes from brainly.com/question/1917026

#learnwithBrainly