Use transformations to graph the following function. Determine the domain, range,
horizontal asymptote, and -intercept of the function.

Which transformation is needed to graph the function ? Choose the
correct answer below.
A. The graph of should be shifted vertically down by 2 units.
C. The graph of should be shifted horizontally left by 2 units.
D. The graph of should be shifted horizontally right by 2 units.

Ask by Peters Clark. in the United States

Nov 04,2024

Question

Use transformations to graph the following function. Determine the domain, range,
horizontal asymptote, and -intercept of the function.

Which transformation is needed to graph the function ? Choose the
correct answer below.
A. The graph of should be shifted vertically down by 2 units.
C. The graph of should be shifted horizontally left by 2 units.
D. The graph of should be shifted horizontally right by 2 units.

Ask by Peters Clark. in the United States

Nov 04,2024

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = 6^x + 2 $$, we can break it down into its components and transformations.

1. **Base Function**: The base function is $$ y = 6^x $$. This is an exponential function that increases rapidly as $$ x $$ increases.

2. **Transformation**: The function $$ f(x) = 6^x + 2 $$ represents a vertical shift of the base function $$ y = 6^x $$ upward by 2 units. Therefore, the correct transformation is:
   - **A. The graph of $$ y = 6^x $$ should be shifted vertically up by 2 units.**

Now, let's determine the domain, range, horizontal asymptote, and $$ y $$-intercept of the function $$ f(x) = 6^x + 2 $$.

### Domain
The domain of $$ f(x) = 6^x + 2 $$ is all real numbers, since the exponential function is defined for all $$ x $$:
$$
\text{Domain: } (-\infty, \infty)
$$

### Range
The range of the base function $$ y = 6^x $$ is $$ (0, \infty) $$. Since we are shifting this graph up by 2 units, the new range becomes:
$$
\text{Range: } (2, \infty)
$$

### Horizontal Asymptote
The horizontal asymptote of the base function $$ y = 6^x $$ is $$ y = 0 $$. After shifting the graph up by 2 units, the horizontal asymptote of $$ f(x) = 6^x + 2 $$ becomes:
$$
\text{Horizontal Asymptote: } y = 2
$$

### $$ y $$-Intercept
To find the $$ y $$-intercept, we evaluate $$ f(0) $$:
$$
f(0) = 6^0 + 2 = 1 + 2 = 3
$$
Thus, the $$ y $$-intercept is:
$$
\text{$$ y $$-intercept: } (0, 3)
$$

### Summary
- **Domain**: $$ (-\infty, \infty) $$
- **Range**: $$ (2, \infty) $$
- **Horizontal Asymptote**: $$ y = 2 $$
- **$$ y $$-Intercept**: $$ (0, 3) $$

So, the correct answer for the transformation needed to graph the function $$ f(x) = 6^x + 2 $$ is:
**A. The graph of $$ y = 6^x $$ should be shifted vertically up by 2 units.**
The graph of $$ y = 6^x $$ should be shifted vertically up by 2 units. The domain is $$ (-\infty, \infty) $$, the range is $$ (2, \infty) $$, the horizontal asymptote is $$ y = 2 $$, and the $$ y $$-intercept is $$ (0, 3) $$.

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