Question

describa la dilatación en cada función en relación con la gráfica de la función principal.

1. ( g(x) = 4x )
2. ( g(x) = 0.8x )
3. ( g(x) = 1.2x )

Explanation:

To solve this, we analyze the vertical dilation of the function \( g(x) \) with respect to the parent function \( f(x) = x \) (the principal linear function). A vertical dilation of a function \( y = a \cdot f(x) \) is determined by the value of \( a \):

• If \( |a| > 1 \), the graph is stretched vertically (narrower).
• If \( 0 < |a| < 1 \), the graph is compressed vertically (wider).

Problem 4: \( g(x) = 4x \)

Step 1: Identify the dilation factor

The function is \( g(x) = 4x \), so the coefficient \( a = 4 \).

Step 2: Analyze the dilation

Since \( |4| = 4 > 1 \), the graph of \( g(x) = 4x \) is a vertical stretch of the principal function \( f(x) = x \) by a factor of 4. This means the graph becomes narrower (steeper) than \( f(x) = x \).

Problem 5: \( g(x) = 0.8x \)

Step 1: Identify the dilation factor

The function is \( g(x) = 0.8x \), so the coefficient \( a = 0.8 \).

Step 2: Analyze the dilation

Since \( 0 < |0.8| = 0.8 < 1 \), the graph of \( g(x) = 0.8x \) is a vertical compression of the principal function \( f(x) = x \) by a factor of \( 0.8 \) (or \( \frac{4}{5} \)). This means the graph becomes wider (less steep) than \( f(x) = x \).

Problem 6: \( g(x) = 1.2x \)

Step 1: Identify the dilation factor

The function is \( g(x) = 1.2x \), so the coefficient \( a = 1.2 \).

Step 2: Analyze the dilation

Since \( |1.2| = 1.2 > 1 \), the graph of \( g(x) = 1.2x \) is a vertical stretch of the principal function \( f(x) = x \) by a factor of \( 1.2 \) (or \( \frac{6}{5} \)). This means the graph becomes narrower (steeper) than \( f(x) = x \).

Resumen de Respuestas:

• Problema 4: Dilatación vertical de estiramiento (stretch) por un factor de 4.
• Problema 5: Dilatación vertical de compresión (compression) por un factor de 0.8.
• Problema 6: Dilatación vertical de estiramiento (stretch) por un factor de 1.2.

Answer:

Step 1: Identify the dilation factor

The function is \( g(x) = 1.2x \), so the coefficient \( a = 1.2 \).

Step 2: Analyze the dilation

Since \( |1.2| = 1.2 > 1 \), the graph of \( g(x) = 1.2x \) is a vertical stretch of the principal function \( f(x) = x \) by a factor of \( 1.2 \) (or \( \frac{6}{5} \)). This means the graph becomes narrower (steeper) than \( f(x) = x \).

Resumen de Respuestas:

• Problema 4: Dilatación vertical de estiramiento (stretch) por un factor de 4.
• Problema 5: Dilatación vertical de compresión (compression) por un factor de 0.8.
• Problema 6: Dilatación vertical de estiramiento (stretch) por un factor de 1.2.