6. Suppose $$ b_{1}(x)=a(x)-5 $$ and $$ b_{2}(x)=a(x-5) $$ for $$ a(x)=x $$ How are the graphs of $$ b_{1} $$ and $$ b_{2} $$ related? Explain.

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UpStudy AI · Accepted Answer

```markdown
### Step 1: Write Down the Given Functions

We are given:
- $$ b_{1}(x) = a(x) - 5 $$
- $$ b_{2}(x) = a(x-5) $$
  
And it is given that:
- $$ a(x) = x $$

---

### Step 2: Substitute $$ a(x) $$ into $$ b_1(x) $$ and $$ b_2(x) $$

**For $$ b_{1}(x) $$:**

$$
b_{1}(x) = a(x) - 5 = x - 5
$$

**For $$ b_{2}(x) $$:**

$$
b_{2}(x) = a(x-5) = (x-5)
$$

---

### Step 3: Compare the Two Functions

Both functions simplify to:

$$
b_{1}(x) = x - 5 \quad \text{and} \quad b_{2}(x) = x - 5
$$

This shows that:

$$
b_{1}(x) = b_{2}(x) \quad \text{for all } x
$$

---

### Step 4: Interpretation of the Graphical Transformations

- **$$ b_{1}(x) = a(x) - 5 $$:**  
  This represents a vertical shift downward by 5 units of the graph of $$ a(x) = x $$.

- **$$ b_{2}(x) = a(x-5) $$:**  
  This represents a horizontal shift to the right by 5 units of the graph of $$ a(x) = x $$.

For the function $$ a(x) = x $$ (a line with slope 1), shifting vertically downward by 5 units and shifting horizontally to the right by 5 units both produce the same linear function, which is why:

$$
x - 5 = (x - 5)
$$

---

### Final Answer

Thus, the graphs of $$ b_{1}(x) $$ and $$ b_{2}(x) $$ are identical. They both represent the line:

$$
y = x - 5
$$
```
The graphs of $$ b_{1}(x) $$ and $$ b_{2}(x) $$ are the same line $$ y = x - 5 $$.

6. Suppose \( b_{1}(x)=a(x)-5 \) and \( b_{2}(x)=a(x-5) \) for \( a(x)=x \) How are the graphs of \( b_{1} \) and \( b_{2} \) related? Explain.

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