Question

analyze the graphs $b(x) = 1.5\sqrt{4 - x} - 1$ and $c(x) = 9\sqrt{4 - x} - 1$. determine the change that occurs from $b(x)$ to $c(x)$. (1 point)
\bigcirc vertical stretch by a factor of 6
\bigcirc vertical compression by a factor of 9
\bigcirc vertical stretch by a factor of 7.5
\bigcirc vertical stretch by a factor of 9

Explanation:

Step1: Recall Vertical Stretch/Compression Rule

For a function \( y = a \cdot f(x) \), if \( |a| > 1 \), it's a vertical stretch by factor \( |a| \); if \( 0 < |a| < 1 \), it's a vertical compression. Here, \( b(x)=1.5\sqrt{4 - x}-1 \) and \( c(x)=9\sqrt{4 - x}-1 \). Let \( f(x)=\sqrt{4 - x}-1 \), then \( b(x)=1.5f(x) \) and \( c(x)=9f(x) \).

Step2: Calculate the Stretch Factor

To find the factor from \( b(x) \) to \( c(x) \), divide the coefficient of \( f(x) \) in \( c(x) \) by that in \( b(x) \). So \( \frac{9}{1.5} = 6 \)? Wait, no—wait, actually, \( b(x)=1.5\sqrt{4 - x}-1 \) can be seen as \( b(x)=1.5(\sqrt{4 - x})-1 \), and \( c(x)=9(\sqrt{4 - x})-1 \). The base function for the square root part is \( \sqrt{4 - x} \), with vertical scaling. The coefficient of \( \sqrt{4 - x} \) in \( b(x) \) is \( 1.5 \), in \( c(x) \) is \( 9 \). The factor of stretch is \( \frac{9}{1.5}=6 \)? Wait, no, wait—wait, maybe I misread. Wait, \( b(x)=1.5\sqrt{4 - x}-1 \), \( c(x)=9\sqrt{4 - x}-1 \). So the transformation from \( b(x) \) to \( c(x) \) is multiplying the coefficient of \( \sqrt{4 - x} \) by \( \frac{9}{1.5}=6 \)? Wait, no, wait: \( b(x) = 1.5 \cdot (\sqrt{4 - x}) - 1 \), \( c(x) = 9 \cdot (\sqrt{4 - x}) - 1 \). So to get from \( b(x) \) to \( c(x) \), we multiply the \( \sqrt{4 - x} \) term by \( \frac{9}{1.5} = 6 \)? Wait, no, that's not right. Wait, actually, the general form for vertical stretch: if \( y = a f(x) \), then from \( y = k f(x) \) to \( y = m f(x) \), the stretch factor is \( \frac{m}{k} \). Here, \( k = 1.5 \), \( m = 9 \), so \( \frac{9}{1.5} = 6 \). Wait, but let's check the options. The first option is vertical stretch by factor 6. Wait, but wait, maybe I made a mistake. Wait, \( b(x) = 1.5\sqrt{4 - x} - 1 \), \( c(x) = 9\sqrt{4 - x} - 1 \). So the part with the square root is scaled by \( \frac{9}{1.5} = 6 \), so it's a vertical stretch by factor 6. Wait, but let's re-express: Let \( u(x) = \sqrt{4 - x} \), then \( b(x) = 1.5u(x) - 1 \), \( c(x) = 9u(x) - 1 \). So to go from \( b(x) \) to \( c(x) \), we have \( c(x) = 6 \cdot (1.5u(x)) - 1 \)? No, wait, \( 9 = 1.5 \times 6 \), so \( c(x) = 6 \times b(x) + 6 \times 1 - 1 \)? No, that's not. Wait, no, the constant term is the same (-1), so the only change is in the coefficient of \( u(x) \). So from \( 1.5u(x) \) to \( 9u(x) \), the factor is \( 9 / 1.5 = 6 \), so vertical stretch by factor 6.

Answer:

A. vertical stretch by a factor of 6