Question

question 1-4
the graph below shows the function $f(x) = x$.
graph of f(x) = x
which function $g(x)$ represents the transformation of $f(x)$ in the graph below?
graph of transformed function
$g(x) = 2f(x)$
$g(x) = \frac{1}{2}f(x)$
$g(x) = f(x) + 2$
$g(x) = f(x) - 2$

Explanation:

Step 1: Analyze the original function

The original function is \( f(x) = x \), which has a slope of 1. The graph of \( f(x) \) is a line passing through the origin with a 45 - degree angle (since for every 1 unit increase in \( x \), \( y \) increases by 1 unit).

Step 2: Analyze the transformation types

• For a vertical stretch or compression: If \( g(x)=a\cdot f(x) \), when \( |a|> 1 \), it is a vertical stretch, and the slope of the line (for a linear function \( f(x)=mx + b \)) will be multiplied by \( a \). When \( 0<|a|<1 \), it is a vertical compression.
• For a vertical shift: If \( g(x)=f(x)+k \), when \( k>0 \), the graph shifts up by \( k \) units, and when \( k < 0 \), the graph shifts down by \( |k| \) units.

Step 3: Analyze each option

• Option 1: \( g(x)=2f(x) \). Since \( f(x)=x \), then \( g(x) = 2x \). The slope of \( g(x) \) is 2, which is steeper than the slope of \( f(x) \) (slope = 1). A steeper slope indicates a vertical stretch (since we are multiplying the function value by 2, which makes the line rise faster for a given \( x \)-increase).
• Option 2: \( g(x)=\frac{1}{2}f(x)=\frac{1}{2}x \). The slope of this line is \( \frac{1}{2} \), which is less steep than the slope of \( f(x) \), so it is a vertical compression, not a steeper line.
• Option 3: \( g(x)=f(x) + 2=x + 2 \). This is a vertical shift up by 2 units. The line will be parallel to \( f(x) \) (same slope) but shifted up, so it will not be steeper.
• Option 4: \( g(x)=f(x)-2=x - 2 \). This is a vertical shift down by 2 units. The line will be parallel to \( f(x) \) (same slope) but shifted down, so it will not be steeper.

From the graph of \( g(x) \), we can see that the line is steeper than the line of \( f(x) \). So the transformation is a vertical stretch by a factor of 2, which corresponds to \( g(x)=2f(x) \).

Answer:

\( g(x) = 2f(x) \) (the first option among the given options for \( g(x) \))