Question

transform the solid black function to match the dotted function.
$y = |x| + 3$
$+ \quad f( x+\square )+ \square$

Explanation:

Step1: Analyze horizontal shift

The original function is \( f(x)=|x| + 3 \) (solid black). The dotted function seems to be shifted left. For a horizontal shift, the transformation is \( f(x + h) \) where \( h>0 \) shifts left by \( h \) units. From the graph, we can see the vertex of the solid function is at \( (0,3) \), and the dotted function's vertex seems to be at \( (-2, 4) \)? Wait, no, let's re - check. Wait, the solid function is \( y = |x|+3 \), so its vertex is at \( (0,3) \). The dotted function: let's see the transformation form \( f(x + a)+b \), where \( f(x)=|x|+3 \). Wait, maybe I made a mistake. Wait, the problem is to transform the solid black function ( \( y = |x|+3 \)) to the dotted function. Let's look at the horizontal shift first. If we have \( f(x + a) \), that is replacing \( x \) with \( x + a \) in \( f(x) \). So \( f(x)=|x| + 3 \), then \( f(x + a)=|x + a|+3 \). Then we add \( b \) to get \( |x + a|+3 + b \).

Looking at the graph, the solid function has vertex at \( (0,3) \), the dotted function: let's see the horizontal shift. If we shift the solid function to the left by 2 units? Wait, no, maybe the other way. Wait, the first box is in \( x+\square \), so if we have \( f(x + a) \), that is a horizontal shift left by \( a \) units (if \( a>0 \)). Wait, maybe the dotted function is \( f(x + 2)+1 \)? Wait, no, let's calculate the vertical and horizontal shifts.

Wait, the solid function is \( y = |x|+3 \). Let's assume the dotted function is obtained by shifting the solid function left by 2 units and up by 1 unit? Wait, no, let's check the vertex. The solid vertex is \( (0,3) \). If we shift left by 2 units, the x - coordinate becomes \( 0-2=-2 \), and up by 1 unit, the y - coordinate becomes \( 3 + 1=4 \). But maybe the dotted function's vertex is at \( (-2,4) \)? Alternatively, maybe I misread the function. Wait, the problem says "Transform the solid black function to match the dotted function". The solid function is \( y = |x|+3 \), and we need to write it in the form \( +f(x+\square)+ \square \). Wait, the first square is for the horizontal shift, the second for the vertical shift.

Wait, maybe the horizontal shift is 2 units to the left (so \( a = 2 \)) and vertical shift is - 2? No, that doesn't make sense. Wait, let's look at the numbers on the graph. The x - axis has - 10, - 9, ..., 0, ..., 10. The y - axis has - 10, ..., 0, ..., 10. The solid function is \( y=|x| + 3 \), so when \( x = 0 \), \( y = 3 \). The dotted function: let's see, if we shift the solid function left by 2 units, then \( f(x+2)=|x + 2|+3 \), and then add 1? Wait, no, maybe the vertical shift is - 2? No, this is getting confusing. Wait, maybe the correct horizontal shift is 2 (left) and vertical shift is - 2? No, let's start over.

The general form of a transformation of a function \( f(x) \) is:

- Horizontal shift: \( f(x + h) \) shifts the graph left by \( h \) units ( \( h>0 \) )
- Vertical shift: \( f(x)+k \) shifts the graph up by \( k \) units ( \( k>0 \) )

We have \( f(x)=|x| + 3 \). Let's assume that the dotted function is obtained by shifting the solid function left by 2 units and up by 1 unit. So \( f(x + 2)=|x + 2|+3 \), then \( f(x + 2)+1=|x + 2|+3 + 1=|x + 2|+4 \). But maybe the dotted function has vertex at \( (-2,4) \), while the solid has vertex at \( (0,3) \). The horizontal shift from \( (0,3) \) to \( (-2,4) \): the horizontal change is \( 0-(-2)=2 \) units to the left, and the vertical change is \( 4 - 3=1 \) unit up. So the horizontal shift is 2 units left (so \( a = 2 \) in \( x + a \)) and vertical shift is 1 unit up (so \( b = 1 \) in \( +b \)).

Wait, but the problem has two boxes: the first in \( x+\square \), the second after the +. So the first box is 2, the second is 1? Wait, maybe I made a mistake. Wait, let's check with the function. If \( f(x)=|x|+3 \), then \( f(x + 2)=|x + 2|+3 \), and then adding 1 gives \( |x + 2|+3 + 1=|x + 2|+4 \). But maybe the dotted function is \( |x + 2|+4 \), which is a transformation of \( |x|+3 \) by shifting left 2 and up 1. So the first box is 2, the second is 1.

Step2: Confirm the shifts

Horizontal shift: To shift the graph of \( y = f(x) \) left by \( a \) units, we use \( y=f(x + a) \). Here, \( a = 2 \), so \( f(x + 2)=|x + 2|+3 \).

Vertical shift: To shift the graph up by \( b \) units, we add \( b \) to the function. So \( f(x + 2)+1=|x + 2|+3 + 1=|x + 2|+4 \), which should match the dotted function.

Answer:

The first box (horizontal shift) is 2, the second box (vertical shift) is 1. So the answer is 2 (in the first box) and 1 (in the second box).