Question
The graph of \( g(x)=x^{3}+1 \) needs to be horizontally compressed by a factor of \( \frac{1}{12} \) What should the value of \( b \) be in the new function \( h(x)=(b x)^{3}+1 \) ? (1 point) \( \frac{1}{12} \) 12 \( -\frac{1}{12} \) -12
Ask by Fuentes Hardy. in the United States
Jan 23,2025
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Answer
The value of \( b \) should be **12**.
Solution
To horizontally compress the graph of \( g(x) = x^3 + 1 \) by a factor of \( \frac{1}{12} \), you need to adjust the function to \( h(x) = (b x)^3 + 1 \).
**Horizontal Compression Factor:**
- A horizontal compression by a factor of \( c \) is achieved by replacing \( x \) with \( b x \) where \( b = \frac{1}{c} \).
- Given \( c = \frac{1}{12} \), we find \( b = \frac{1}{\frac{1}{12}} = 12 \).
So, the value of \( b \) should be **12**.
**Answer:** 12
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Simplify this solution Extra Insights
To horizontally compress the function \( g(x) = x^3 + 1 \) by a factor of \( \frac{1}{12} \), you need to replace \( x \) in the original function with \( \frac{x}{\frac{1}{12}} \) (or \( 12x \)). This leads to the new function form being \( h(x) = (12x)^3 + 1 \). Therefore, the value of \( b \) should be 12. Additionally, when dealing with transformations of functions, remember that a horizontal compression by a factor of \( k \) can be represented by multiplying the input \( x \) by \( \frac{1}{k} \). This means that if you're compressing by \( \frac{1}{12} \), you'd effectively be using a multiplication factor of 12 in the new function, which matches our earlier finding!
