Question

the graph of \\(g(x)\\) is shown.

the graph of \\(h(x) = g(x + 5)\\) has choose an answer and choose an answer.

Explanation:

Identify asymptotes of the original function

From the given graph of \(g(x)\):

• The vertical asymptote is at \(x = -2\).
• The horizontal asymptote is at \(y = 3\).

Apply the horizontal translation

The transformed function is \(h(x) = g(x + 5)\).

• Replacing \(x\) with \(x + 5\) shifts the entire graph horizontally to the left by 5 units.
• The new vertical asymptote is:

$$ x = -2 - 5 = -7 $$

• The horizontal asymptote remains unchanged:

$$ y = 3 $$

Determine the final properties

The graph of \(h(x) = g(x + 5)\) has:

• A vertical asymptote at \(x = -7\)
• A horizontal asymptote at \(y = 3\)

Answer:

The graph of \(h(x) = g(x + 5)\) has <blank>a vertical asymptote at \(x = -7\)</blank> and <blank>a horizontal asymptote at \(y = 3\)</blank>.