Question

graph the function $f(x) = 2\log_4(x + 5)$ on the axes below. you must plot the asymptote and any two points with integer coordinates.
asymptote:
vertical
horizontal

Explanation:

Step1: Find the vertical asymptote

For a logarithmic function of the form \( y = a\log_b(x - h)+k \), the vertical asymptote is at \( x = h \). In the function \( f(x)=2\log_4(x + 5) \), we can rewrite it as \( f(x)=2\log_4(x-(- 5))+0 \), so \( h=-5 \). Thus, the vertical asymptote is \( x=-5 \).

Step2: Find two points with integer coordinates

• Let's find the point when \( x + 5 = 1 \) (since \( \log_b(1)=0 \) for any \( b>0,b

eq1 \)). Solving \( x + 5=1 \), we get \( x=1 - 5=-4 \). Then \( f(-4)=2\log_4(1)=2\times0 = 0 \). So the point is \( (-4,0) \).

• Let's find the point when \( x + 5 = 4 \) (since \( \log_4(4) = 1 \)). Solving \( x+5 = 4 \), we get \( x=4 - 5=-1 \). Then \( f(-1)=2\log_4(4)=2\times1 = 2 \). So the point is \( (-1,2) \).

Answer:

The vertical asymptote is \( x = - 5 \). Two points with integer coordinates are \( (-4,0) \) and \( (-1,2) \) (you can plot these on the graph along with the vertical asymptote \( x=-5 \)).