Question

which graph shows \\(f(x) = 3^{x+4}\\) and the translation \\(g(x) = f(x-5)\\)?

Explanation:

Analyze the original function

The original function is given as:

$$f(x) = 3^{x+4}$$

Let's evaluate some key points for \(f(x)\):

• When \(x = -4\), \(f(-4) = 3^{-4+4} = 3^0 = 1\).
• When \(x = -3\), \(f(-3) = 3^{-3+4} = 3^1 = 3\).
• When \(x = -2\), \(f(-2) = 3^{-2+4} = 3^2 = 9\).

Looking at the blue curve labeled \(f(x)\) in the image:

• At \(x = -4\), the curve is at \(y = 1\).
• At \(x = -3\), the curve is at \(y = 3\).
• At \(x = -2\), the curve is at \(y = 9\).

This matches the blue curve perfectly.

Analyze the transformation

The transformed function is defined as:

$$g(x) = f(x-5)$$

Substituting \(x-5\) into \(f(x)\):

$$g(x) = 3^{(x-5)+4} = 3^{x-1}$$

This represents a horizontal translation of the graph of \(f(x)\) to the right by \(5\) units.

Evaluate key points for the translation

Let's find the corresponding points for \(g(x)\):

• The point \((-4, 1)\) on \(f(x)\) shifts \(5\) units to the right to become \((-4+5, 1) = (1, 1)\).
• The point \((-3, 3)\) on \(f(x)\) shifts \(5\) units to the right to become \((-3+5, 3) = (2, 3)\).
• The point \((-2, 9)\) on \(f(x)\) shifts \(5\) units to the right to become \((-2+5, 9) = (3, 9)\).

Compare with the provided graph

In the provided image:

• The red curve labeled \(g(x)\) is shifted vertically upwards. For example, its horizontal asymptote is at \(y = 5\) instead of \(y = 0\).
• Since \(g(x) = f(x-5)\) is a pure horizontal shift of \(5\) units to the right, its horizontal asymptote must remain at \(y = 0\).
• Therefore, the graph shown in the image is incorrect because the red curve represents a vertical shift rather than a horizontal shift.

Determine the correct graph characteristics

The correct graph must show:

• The blue curve \(f(x) = 3^{x+4}\) passing through \((-4, 1)\), \((-3, 3)\), and \((-2, 9)\).
• The red curve \(g(x) = 3^{x-1}\) passing through \((1, 1)\), \((2, 3)\), and \((3, 9)\), with both curves sharing the horizontal asymptote \(y = 0\).

Answer:

The graph shown in the image is incorrect because the red curve \(g(x)\) represents a vertical shift instead of a horizontal shift.

The correct graph must show:

• The blue curve \(f(x) = 3^{x+4}\) passing through the points \((-4, 1)\), \((-3, 3)\), and \((-2, 9)\).
• The red curve \(g(x) = f(x-5) = 3^{x-1}\) shifted horizontally to the right by \(5\) units, passing through the points \((1, 1)\), \((2, 3)\), and \((3, 9)\), with both curves sharing the horizontal asymptote \(y = 0\).