Question

which graph represents the function $f(x) = |x + 3|$?

Explanation:

Step1: Recall the vertex form of absolute value function

The general form of an absolute value function is \( f(x) = |x - h| + k \), where \((h, k)\) is the vertex of the V - shaped graph. For the function \( f(x)=|x + 3|\), we can rewrite it as \( f(x)=|x-(- 3)|+0 \). So, the vertex of the graph should be at \((-3,0)\). Also, the absolute value function \( y = |x| \) has a V - shape with the vertex at the origin \((0,0)\) and opens upwards. The function \( y=|x + 3|\) is a horizontal shift of the parent function \( y = |x| \) 3 units to the left.

Step2: Analyze the given graphs

• The first graph (the V - shaped one) in the image: Let's check the vertex. Wait, in the first graph shown, the vertex is at \((0,3)\)? No, wait, maybe there is a mis - look. Wait, the correct vertex for \( f(x)=|x + 3| \) should be at \( x=-3,y = 0 \). But maybe the user's image has some other graphs. Wait, the second graph is a linear graph (a straight line with positive slope), but the absolute value function is a piece - wise linear function with a V - shape. So the graph that is V - shaped (the first one in the provided image, even if the vertex seems mis - placed in the initial analysis, but the second graph is a straight line which can't be an absolute value graph as absolute value graphs are V - shaped) should be the one representing \( f(x)=|x + 3| \). The absolute value function \( y = |x + 3| \) will have a V - shape, with the left side having a slope of - 1 and the right side having a slope of 1, and the vertex at \( (-3,0) \). The second graph is a linear function (non - piecewise), so it can't represent an absolute value function. So the V - shaped graph (the first one) is the correct representation.

Answer:

The V - shaped graph (the first graph in the given image) represents the function \( f(x)=|x + 3| \). (Assuming the first graph is the V - shaped one and the second is a straight line. If we consider the standard form, the graph of \( y = |x+3| \) is a V - shaped graph with vertex at \((-3,0)\), opening upwards, and having two linear segments: \( y=-(x + 3)=-x - 3 \) for \( x<-3 \) and \( y=x + 3 \) for \( x\geq - 3 \))