Question

function g can be thought of as a scaled version of f(x)=|x|. what is the equation for g(x)? choose 1 answer: a ( g(x) = \frac{3}{2}|x| ) b ( g(x) = -\frac{2}{3}|x| ) c ( g(x) = |x - 3| - 2 ) d ( g(x) = |x - 2| + 3 ) (with a graph showing functions f and g)

Explanation:

Step1: Analyze the parent function and transformation

The parent function is \( f(x) = |x| \), which has a V - shape with vertex at the origin. The graph of \( g(x) \) is a scaled and possibly reflected version of \( f(x) \). Let's consider the slope of the lines. For \( f(x)=|x| \), when \( x>0 \), \( f(x)=x \) (slope \( m = 1 \)), when \( x < 0 \), \( f(x)=-x \) (slope \( m=- 1 \)).

Looking at the graph of \( g(x) \) (the dashed line), let's take a point. For example, when \( x = 3 \), let's find the value of \( y \). From the graph, we can see that the line of \( g(x) \) (dashed) has a slope. Let's check the options:

• Option A: \( g(x)=\frac{3}{2}|x| \). The slope for \( x>0 \) would be \( \frac{3}{2} \), which is greater than 1, so the line would be steeper than \( f(x) \), but our dashed line seems less steep or has a negative slope? Wait, no, wait the dashed line: let's check the direction. The dashed line has a negative slope (going down from left to right). So it's a reflection over the x - axis (since the slope is negative) and a vertical scaling.

• Option B: \( g(x)=-\frac{2}{3}|x| \). When \( x>0 \), \( g(x)=-\frac{2}{3}x \) (slope \( m =-\frac{2}{3} \)), when \( x < 0 \), \( g(x)=-\frac{2}{3}(-x)=\frac{2}{3}x \) (slope \( m=\frac{2}{3} \)). Let's check a point. Let's take \( x = 3 \), then \( g(3)=-\frac{2}{3}\times3=- 2 \). Wait, maybe better to check the slope. The original \( f(x) \) has slope 1 (for \( x>0 \)). The dashed line: let's see, when \( x = 3 \), what's \( y \)? From the graph, if we assume the grid is 1 unit per square, when \( x = 3 \), the \( y \)-value for \( g(x) \) is - 2? Wait, no, let's think again.

Wait, the function \( f(x)=|x| \) has a vertex at (0,0). The graph of \( g(x) \): let's check the slope. For a linear function of the form \( y = a|x| \), the slope of the right - hand side (x>0) is \( a \) (if \( a>0 \)) or \( -a \) (if \( a < 0 \)).

Looking at the dashed line ( \( g(x) \) ), when \( x = 3 \), let's find \( y \). If we take two points on \( g(x) \): let's say \( (3,- 2) \) and \( (0,0) \). The slope between (0,0) and (3, - 2) is \( m=\frac{-2 - 0}{3-0}=-\frac{2}{3} \). So the equation of the line for \( x>0 \) would be \( y=-\frac{2}{3}x \), and for \( x < 0 \), \( y=\frac{2}{3}x \) (since \( |x|=-x \) when \( x < 0 \), so \( y =-\frac{2}{3}(-x)=\frac{2}{3}x \)). So \( g(x)=-\frac{2}{3}|x| \), which is option B.

• Option C: \( g(x)=|x - 3|-2 \). This is a horizontal shift (3 units right) and vertical shift (2 units down). But the vertex of \( g(x) \) in this case would be at (3, - 2), but our dashed line has vertex at (0,0), so this is not correct.

• Option D: \( g(x)=|x - 2|+3 \). Vertex at (2,3), which is not the case for our dashed line (vertex at (0,0)), so this is incorrect.

Step2: Eliminate other options

• Option C and D are incorrect because they are horizontal and vertical shifts (vertex not at origin), while our \( g(x) \) has vertex at the origin (same as \( f(x) \)'s vertex, just scaled and reflected).

• Option A: \( \frac{3}{2}|x| \) has a positive slope for \( x>0 \), but our dashed line has a negative slope, so A is incorrect.

So the correct option is B.

Answer:

B. \( g(x)=-\frac{2}{3}|x| \)