Question

Analyze the effect of the leading coefficient on j(x)=2513x−−−√+4
in comparison to k(x)=13x−−−√+4
. Enter your response in fraction form if needed.(1 point)
a vertical compression by a factor of

Answer 1

To analyze the effect of the leading coefficient on the functions \( j(x) = 2513\sqrt{x} + 4 \) and \( k(x) = 13\sqrt{x} + 4 \), we will first examine what happens when the leading coefficient increases.

The leading coefficient affects the vertical stretching and compressing of the graph of a function.

1. **Identifying the leading coefficients**:
- For \( j(x) \):
- The leading coefficient is \( 2513 \).
- For \( k(x) \):
- The leading coefficient is \( 13 \).

2. **Comparing the leading coefficients**:
- To find out how much \( j(x) \) is compressed or stretched compared to \( k(x) \), we simply take the ratio of the leading coefficients:
\[
\frac{\text{leading coefficient of } j(x)}{\text{leading coefficient of } k(x)} = \frac{2513}{13}
\]

3. **Calculating the ratio**:
- We can simplify this ratio:
\[
\frac{2513}{13} = 193
\]

4. **Interpreting the result**:
- Since \( j(x) \) has a leading coefficient that is larger than that of \( k(x) \), the graph of \( j(x) \) is stretched vertically by a factor of \( 193 \) compared to \( k(x) \). This means \( k(x) \) undergoes a vertical compression when comparing to \( j(x) \).

To express it in terms of compression, we note that a larger leading coefficient results in a smaller output for the same input \( x \), signifying compression.

5. **Conclusion:**
Since compression is the inverse of stretching, we thus find:
\[
\text{Vertical compression factor} = \frac{1}{193}
\]

Therefore, the effect of the leading coefficient on \( j(x) \) compared to \( k(x) \) results in a vertical compression by a factor of \( \frac{1}{193} \).