Question

in exercises 1–4, describe the transformation of f represented by g. then graph each function. (see example 1.)

1. $f(x) = x^4$, $g(x) = x^4 + 3$
2. $f(x) = x^4$, $g(x) = (x - 5)^4$
3. $f(x) = x^5$, $g(x) = (x - 2)^5 - 1$
4. $f(x) = x^6$, $g(x) = (x + 1)^6 - 4$

reasoning in exercises 5–8, match the function with the correct transformation of the graph of f. explain your reasoning.

1. $y = f(x - 2)$
2. $y = f(x + 2) + 2$
3. $y = f(x - 2) + 2$
4. $y = f(x) - 2$

(graphs for options a, b, c, d are shown with various polynomial graphs.)

Explanation:

Step1: Analyze the transformation rules

For a function \( y = f(x - h)+k \), the graph of \( f(x) \) is shifted \( h \) units horizontally (right if \( h>0 \), left if \( h < 0 \)) and \( k \) units vertically (up if \( k>0 \), down if \( k < 0 \)).

Step2: Apply to \( y = f(x + 2)+2 \)

Rewrite \( x + 2\) as \( x-(- 2) \). So \( h=-2\) and \( k = 2\).
- Horizontal shift: Since \( h=-2<0 \), the graph of \( f(x) \) is shifted 2 units to the left.
- Vertical shift: Since \( k = 2>0 \), the graph of \( f(x) \) is shifted 2 units up.

Now, to match with the graphs:
- The original graph of \( f(x) \) is symmetric about the \( y \)-axis (even function, as seen from the given \( f(x) \) graph).
- After shifting left 2 units and up 2 units, we need to find the graph that shows this transformation. Looking at the options, we analyze the position of the vertex (or key points). The original graph has its minimum (or key point) at the origin. After shifting left 2 and up 2, the key point moves to \((-2,2)\). We check the graphs:
- Graph A: Shifted down? No.
- Graph B: Shifted right? No.
- Graph C: Shifted left? Let's see the position. Wait, maybe I made a mistake. Wait, the original \( f(x) \) graph (the blue one) has a shape with a minimum at the origin. For \( y=f(x + 2)+2 \), replacing \( x \) with \( x + 2 \) shifts left 2, and adding 2 shifts up 2. So the new graph should have its key point at \((-2,2)\). Let's check the graphs:
Looking at the options, let's assume the graphs are labeled A, B, C, D. Wait, the user provided graphs:
- The original \( f(x) \) graph is on the top right (symmetric about y-axis).
- For \( y = f(x + 2)+2 \), the transformation is left 2, up 2. So we need to find the graph that is shifted left 2 and up 2 from the original.

Wait, maybe the correct graph is the one that has the vertex at \((-2,2)\). Let's re - evaluate.

Alternatively, let's consider the transformation steps again. The rule for horizontal shift: \( f(x + c) \) is shift left \( c \) units, \( f(x - c) \) is shift right \( c \) units. Vertical shift: \( f(x)+k \) is shift up \( k \) units, \( f(x)-k \) is shift down \( k \) units.

For \( y=f(x + 2)+2 \), it's \( f(x-(-2))+2 \), so left 2, up 2.

Now, looking at the graphs:
- The original \( f(x) \) has a minimum at \((0,0)\). After transformation, the minimum should be at \((-2,2)\).

Looking at the given graphs (A: shifted down, B: shifted right, C: shifted left? Wait, maybe the correct graph is the one that is shifted left 2 and up 2. Let's assume the correct graph is the one that shows this. Wait, maybe I need to re - check.

Wait, the problem is to match \( y = f(x + 2)+2 \) with the correct transformation graph.

After applying the transformation rules (left 2, up 2), we can conclude the correct graph (but since the user's graph labels are a bit unclear, but based on the transformation, the answer for the match is the graph that is shifted 2 units left and 2 units up. If we assume the options are as per the standard, and after analysis, the correct graph (let's say) is the one that is shifted left 2 and up 2. But maybe the user's graph for this problem (question 6) has the correct graph as, for example, if we consider the graphs:

Wait, the user's image has:

- For question 6: \( y = f(x + 2)+2 \), and the original \( f(x) \) graph (top right) is symmetric about y - axis.

After shifting left 2 (so the center of symmetry moves to \( x=-2 \)) and up 2 (so the y - coordinate of the center moves to \( y = 2 \)). So the correct graph should have its center (or key point) at \((-2,2)\).

Assuming the graphs are labeled, and after analysis, the correct match is the graph that shows this shift. But since the user's graph options are A, B, C, D (from the bottom: A, B, C, D), let's re - check:

- Graph A: Shifted down? No.
- Graph B: Shifted right? No.
- Graph C: Let's see, the position. Maybe the correct graph is the one that is shifted left 2 and up 2. But maybe I made a mistake. Wait, the key is the transformation: left 2, up 2. So the answer for the match (question 6) is the graph that represents this shift. But since the user is asking for the solution, let's focus on the transformation description first.

Answer:

The transformation of \( f \) represented by \( y = f(x + 2)+2 \) is a horizontal shift 2 units to the left and a vertical shift 2 units up. To match with the graph, the correct graph (assuming the options) is the one that shows this shift (e.g., if the graphs are labeled, the graph with the key point at \((-2,2)\)). But based on the transformation rules, the transformation is a horizontal shift 2 units left and vertical shift 2 units up.