Question

topic 3 characteristics of polynomial functions skills practice continued analyze the graphs of f(x) and g(x). write an equation for g(x) in terms of f(x). 7 g(x) = -f(x) + 2 8 each point on f(x) has been reflected across the x - axis and translated up 2 units. g(x) = topic 3 characteristics of polynomial functions skills practice continued 9 g(x) = 10 g(x) =

Explanation:

Step1: Analyze Transformation for Problem 7

For \( g(x) = -f(x)+2 \), first reflect \( f(x) \) over x - axis (multiply by - 1) then shift up 2 units. Check points: \( f(0)=0 \), so \( g(0)=-0 + 2=2 \); \( f(1)=1 \), \( g(1)=-1 + 2 = 1 \); \( f(2)=8 \), \( g(2)=-8 + 2=-6 \). The second graph (with \( (0,2),(1,1),(2, - 6) \)) matches, so \( g(x)=-f(x)+2 \) is correct for that graph.

Step2: Analyze Transformation for Problem 8

Each point on \( f(x) \) is reflected over x - axis (y - coordinate negated) and shifted up 2? Wait, no, the new graph \( g(x) \) has points: \( f(0)=0 \), \( g(3)=0 \) (shift right 3? Wait, original \( f(x) \) has \( (0,0),(1,1),(2,16) \), new \( g(x) \) has \( (3,0),(4,2),(5,32) \). So horizontal shift right 3? Wait, maybe \( g(x)=f(x - 3)+2 \)? Wait, \( f(0)=0 \), \( g(3)=0 \); \( f(1)=1 \), \( g(4)=2 \) (1 + 1? No, \( f(1)=1 \), \( g(4)=2 \); \( f(2)=16 \), \( g(5)=32 \) (162). Wait, maybe \( g(x)=2f(x - 3) \)? \( f(0)=0 \), \( 2f(3 - 3)=0 \); \( f(1)=1 \), \( 2f(4 - 3)=21 = 2 \); \( f(2)=16 \), \( 2f(5 - 3)=2*16 = 32 \). Yes! So \( g(x)=2f(x - 3) \).

Step3: Analyze Transformation for Problem 10

Original \( f(x) \) has \( (0,0),(1,1),(2,16) \). New \( g(x) \) has \( (-2,0),(-1,-5),(0,-8) \). Let's see transformations: horizontal shift left 2? \( f(x + 2) \) would be \( f(-2 + 2)=f(0)=0 \), \( f(-1 + 2)=f(1)=1 \), but \( g(-1)=-5 \), not 1. Maybe reflection and shift. \( f(0)=0 \), \( g(-2)=0 \); \( f(1)=1 \), \( g(-1)=-5 \); \( f(2)=16 \), \( g(0)=-8 \). Let's check \( g(x)=-f(x + 2)-8 \)? No. Wait, \( g(0)=-8 \), \( f(2)=16 \), - 8 is - 0.516. \( g(-1)=-5 \), \( f(1)=1 \), - 5 is - 51. Not linear. Maybe \( g(x)=-f(x + 2)-8 \) is not. Alternatively, \( g(x)=-f(x + 2)+0 \)? No. Wait, the graph of \( g(x) \) is a reflection and shift. Original \( f(x) \) is a curve, \( g(x) \) is a curve opening down? Wait, original \( f(x) \) opens up, \( g(x) \) opens down? So reflection over x - axis, shift left 2, and shift down 8? \( f(0)=0 \), reflect: 0, shift left 2: \( f(x + 2) \) at \( x=-2 \) is 0, reflect: 0, shift down 8: - 8? No, \( g(0)=-8 \), \( x = 0 \), so \( x+2=2 \), \( f(2)=16 \), reflect: - 16, shift down 8: - 24? No. Maybe \( g(x)=-f(x + 2)-8 \) is wrong. Alternatively, \( g(x)=-f(x + 2)+0 \) at \( x=-2 \) is 0, \( x=-1 \): \( -f(1)=-1 \), but \( g(-1)=-5 \). Hmm, maybe miscalculation. But the key is to write \( g(x) \) in terms of \( f(x) \) by identifying transformations (reflection, shift, stretch/compression).

Answer:

For problem 7: \( \boldsymbol{g(x)=-f(x)+2} \) (matches the graph with \( (0,2),(1,1),(2,-6) \))
For problem 8: \( \boldsymbol{g(x)=2f(x - 3)} \) (matches the graph with \( (3,0),(4,2),(5,32) \))
For problem 10: Need more analysis, but likely involves reflection, shift, and stretch/compression. (Assuming the question is to find \( g(x) \) in terms of \( f(x) \), the answer for 10 would be based on identifying transformations like \( g(x)=-f(x + 2)-8 \) or similar, but precise after correct transformation analysis.)