Question

what kind of transformation converts the graph of $f(x) = -9(x - 1)^{2} - 7$ into the graph of $g(x) = -(x - 3)^{2} - 7$? horizontal stretch vertical shrink vertical stretch horizontal shrink

Explanation:

Step1: Recall transformation rules

For a quadratic function in the form \( y = a(x - h)^2 + k \), the coefficient \( a \) affects vertical stretching/shrinking, and changes in \( h \) affect horizontal shifts. Here, we focus on the coefficient \( a \) for vertical transformations and the horizontal/vertical stretch/shrink rules. A vertical stretch/shrink is determined by the absolute value of \( a \): if \( |a_1| > |a_2| \), it's a vertical shrink from \( f \) to \( g \), and vice versa for stretch. For horizontal stretch/shrink, the transformation involves the coefficient inside the parentheses, but here we look at the vertical coefficient first.

Step2: Analyze coefficients of \( f(x) \) and \( g(x) \)

For \( f(x) = -9(x - 1)^2 - 7 \), the coefficient \( a_f = -9 \), so \( |a_f| = 9 \). For \( g(x) = -(x - 3)^2 - 7 \), the coefficient \( a_g = -1 \), so \( |a_g| = 1 \). Since \( |a_g|=\frac{|a_f|}{9} \) (because \( 9\times\frac{1}{9}=1 \)), this means we are reducing the vertical stretch factor from 9 to 1, which is a vertical shrink (since we're making the graph "flatter" vertically, reducing the vertical scaling factor). Wait, no—wait, vertical stretch/shrink: if \( |a| > 1 \), it's a vertical stretch; if \( 0 < |a| < 1 \), it's a vertical shrink. So from \( |a| = 9 \) (stretched) to \( |a| = 1 \) (less stretched, i.e., shrunk vertically). Wait, but also, let's check horizontal: no, the horizontal shift is from \( x - 1 \) to \( x - 3 \), which is a horizontal shift, not stretch/shrink. The stretch/shrink is about the coefficient of the squared term. Wait, the question is about the transformation from \( f \) to \( g \). Let's write \( g(x) \) in terms of \( f(x) \)'s structure. Let's see: \( f(x) = -9(x - 1)^2 - 7 \), \( g(x) = -1(x - 3)^2 - 7 \). To get from \( f \) to \( g \), we can think of changing the coefficient from -9 to -1 (vertical) and the horizontal shift, but the options are about stretch/shrink. Wait, maybe I made a mistake. Wait, vertical stretch/shrink: when you have \( y = a f(x) \), vertical stretch by factor \( |a| \) if \( |a| > 1 \), shrink if \( 0 < |a| < 1 \). Here, \( f(x) \) has a vertical stretch factor of 9, \( g(x) \) has 1. So to go from \( f(x) \) to \( g(x) \), we divide the function by 9 (since \( -9(x - 1)^2 - 7 \) to \( -(x - 3)^2 - 7 \) is like multiplying \( f(x) \) by \( \frac{1}{9} \) in the vertical direction? Wait, no, the horizontal stretch/shrink is when the transformation is inside the function, like \( f(kx) \) for horizontal. Wait, no, the standard form for horizontal stretch/shrink is \( y = f(kx) \), which is a horizontal stretch by \( \frac{1}{k} \) if \( 0 < k < 1 \), shrink if \( k > 1 \). But here, the function is \( (x - h) \), so the horizontal stretch/shrink would be if we have \( (kx - h) \), but here we have \( (x - 1) \) to \( (x - 3) \), which is a shift, not a stretch. Wait, maybe the question is about the vertical coefficient. Wait, the options are horizontal stretch, vertical shrink, vertical stretch, horizontal shrink. Wait, let's re-express:

Wait, \( f(x) = -9(x - 1)^2 - 7 \), \( g(x) = -1(x - 3)^2 - 7 \). Let's factor out the coefficient of the squared term. To get from \( f(x) \) to \( g(x) \), we can write \( g(x) = \frac{1}{9} \times [-9(x - 3)^2 - 63] \)? No, that's not right. Wait, maybe the horizontal stretch/shrink is when the coefficient inside the parentheses changes, but here the inside is \( (x - 1) \) to \( (x - 3) \), which is a shift. Wait, no, the stretch/shrink is about the coefficient of the \( (x - h) \) term? No, the standard form for a quadratic is \( y = a(x - h)^2 + k \), where \( a \) is vertical stretch/shrink, and horizontal stretch/shrink would be if you have \( y = a(b(x - h))^2 + k \), so \( b \) is the horizontal stretch factor (stretch by \( 1/b \) if \( b > 1 \), shrink by \( b \) if \( 0 < b < 1 \)). But in our case, \( f(x) = -9(x - 1)^2 - 7 \) (so \( b = 1 \) for \( f \)), \( g(x) = -1(x - 3)^2 - 7 \) ( \( b = 1 \) for \( g \) ). Wait, that can't be. Wait, maybe the problem is that I misread the functions. Wait, no, the functions are \( f(x) = -9(x - 1)^2 - 7 \) and \( g(x) = -(x - 3)^2 - 7 \). So the difference is in the coefficient of the squared term (vertical) and the horizontal shift. But the options are about stretch/shrink, not shift. So let's focus on the vertical coefficient. From \( a = -9 \) to \( a = -1 \): the absolute value goes from 9 to 1, which is a vertical shrink (since we're reducing the vertical stretch factor; a vertical shrink is when \( |a| \) decreases from a larger value to a smaller one, making the graph less "tall" vertically). Wait, but also, is there a horizontal stretch/shrink? No, because the horizontal part is \( (x - 1) \) to \( (x - 3) \), which is a shift, not a stretch. So the transformation is a vertical shrink? Wait, no, wait: vertical stretch is when \( |a| > 1 \), vertical shrink when \( 0 < |a| < 1 \). So if we go from \( |a| = 9 \) (stretched) to \( |a| = 1 \) (not stretched as much, i.e., shrunk vertically). So the correct answer should be vertical shrink? Wait, but let's check the options. The options are horizontal stretch, vertical shrink, vertical stretch, horizontal shrink. Wait, maybe I made a mistake in the direction. Wait, let's think of \( g(x) \) as a transformation of \( f(x) \). Let's write \( g(x) = \frac{1}{9} \times [-9(x - 3)^2 - 7 \times 9] \)? No, that's not helpful. Wait, another approach: the general form for vertical stretch/shrink is \( y = c \cdot f(x) \), where \( c \) is the vertical stretch factor. If \( c > 1 \), vertical stretch; if \( 0 < c < 1 \), vertical shrink. Here, \( g(x) = \frac{1}{9} \cdot [-9(x - 3)^2 - 7 \times 9] \)? No, that's not matching. Wait, maybe the horizontal stretch/shrink is when the coefficient inside the function changes. Wait, the function \( f(x) = -9(x - 1)^2 - 7 \), to get \( g(x) = -(x - 3)^2 - 7 \), we can see that the \( (x - 1) \) becomes \( (x - 3) \) (horizontal shift) and the coefficient -9 becomes -1 (vertical change). But the question is about stretch/shrink, not shift. So the stretch/shrink is about the coefficient of the squared term. So from \( -9(x - 1)^2 \) to \( -1(x - 3)^2 \), the vertical coefficient goes from 9 to 1 (absolute value), so it's a vertical shrink (since we're reducing the vertical stretch factor). Wait, but let's confirm: vertical shrink is when the graph is compressed vertically, i.e., the \( y \)-values are multiplied by a factor between 0 and 1. Here, multiplying the \( y \)-values of \( f(x) \) by \( \frac{1}{9} \) (since \( -9 \times \frac{1}{9} = -1 \)) would give \( g(x) \) in terms of the \( (x - 3) \) part, but there's also a horizontal shift. But the options don't include shift, so we have to consider that maybe the horizontal shift is not part of the stretch/shrink, and the main change in the coefficient is vertical. Wait, but maybe I messed up horizontal vs vertical. Wait, horizontal stretch/shrink is when the transformation is inside the function, like \( f(kx) \), which is a horizontal stretch by \( 1/k \) if \( k < 1 \), shrink by \( k \) if \( k > 1 \). But in our case, the function is \( (x - h) \), so the horizontal component is linear in \( x \) with coefficient 1. So to get from \( (x - 1) \) to \( (x - 3) \), it's a shift, not a stretch. So the only stretch/shrink is vertical, from \( |a| = 9 \) to \( |a| = 1 \), which is a vertical shrink (because we're making the graph less stretched vertically, i.e., shrinking it vertically). Wait, but let's check the options again. The options are:

- horizontal stretch

- vertical shrink

- vertical stretch

- horizontal shrink

So the correct answer should be vertical shrink? Wait, no, wait: when \( |a| \) decreases, it's a vertical shrink. For example, if you have \( y = 2x^2 \) (stretched) to \( y = x^2 \) (shrunk vertically), because the graph becomes less tall. So yes, from \( |a| = 9 \) to \( |a| = 1 \), it's a vertical shrink.

Wait, but maybe I made a mistake. Let's re-express \( g(x) \) as a transformation of \( f(x) \). Let's see: \( f(x) = -9(x - 1)^2 - 7 \), \( g(x) = -1(x - 3)^2 - 7 \). Let's factor out the -9 from \( f(x) \): \( f(x) = -9[(x - 1)^2] - 7 \). To get \( g(x) \), we can write \( g(x) = \frac{1}{9} \times [-9(x - 3)^2] - 7 \), but that would involve a horizontal shift and a vertical shrink. Wait, but the horizontal shift is separate. However, the question is about the transformation that converts \( f \) to \( g \), considering the stretch/shrink. Since the vertical coefficient goes from 9 to 1 (absolute value), it's a vertical shrink. So the answer is vertical shrink.

Answer:

vertical shrink (the option labeled "vertical shrink")