Question

drag each label to the correct location on the table. each label can be used more than once. match the attributes to the quadratic functions. axis of symmetry: ( x = 1 ) minimum value: -1 x-intercept: ( (2,0) ) y-intercept: ( (0,8) ) x-intercept: ( (-2,0) ) y-intercept: ( (0,-8) ) ( f(x) = x^2 - 2x - 8 ) ( g(x) = x^2 + 6x + 8 ) ( h(x) = -x^2 + 2x )

Answer:

for \( f(x) = x^2 - 2x - 8 \)

Step 1: Find y - intercept

Set \( x = 0 \), then \( f(0)=0^{2}-2\times0 - 8=-8 \). So y - intercept is \( (0, - 8) \).

Step 2: Find x - intercept

Set \( f(x)=0 \), i.e., \( x^{2}-2x - 8 = 0 \). Factor the quadratic: \( (x - 4)(x+2)=0 \), so \( x = 4 \) or \( x=-2 \)? Wait, wait, let's check again. Wait, the given x - intercept option has \( (2,0) \)? Wait, maybe I made a mistake. Wait, \( x^{2}-2x - 8=(x - 4)(x + 2)=0 \), solutions \( x = 4 \) and \( x=-2 \). But the option has \( (2,0) \)? Wait, maybe the function is \( f(x)=x^{2}-2x - 8 \), let's complete the square. \( f(x)=x^{2}-2x+1 - 1 - 8=(x - 1)^{2}-9 \). So axis of symmetry is \( x = 1 \), minimum value is \( - 9 \)? Wait, the given minimum value is \( - 1 \). Wait, maybe I misread the function. Wait, the user's function for \( f(x) \) is \( x^{2}-2x - 8 \)? Wait, the other function \( h(x)=-x^{2}+2x \), let's check \( h(x) \). \( h(x)=-x^{2}+2x=-(x^{2}-2x)=-(x^{2}-2x + 1-1)=-(x - 1)^{2}+1 \), so axis of symmetry \( x = 1 \), maximum value 1? But the minimum value given is \( - 1 \). Wait, maybe the function \( f(x) \) is \( x^{2}-2x - 8 \), let's re - evaluate.

Wait, let's match with the given attributes:

1. For \( f(x)=x^{2}-2x - 8 \):

• Y - intercept: when \( x = 0 \), \( y=0 - 0 - 8=-8 \), so y - intercept \( (0,-8) \).
• X - intercept: solve \( x^{2}-2x - 8 = 0\), \( (x - 4)(x + 2)=0\), \( x = 4 \) or \( x=-2 \). But the given x - intercepts are \( (2,0) \) and \( (-2,0) \). Wait, maybe the function is \( f(x)=x^{2}-2x - 8 \), let's check \( x = 2 \): \( f(2)=4-4 - 8=-8

eq0 \). So maybe the function is \( f(x)=x^{2}-2x - 8 \), axis of symmetry \( x=-\frac{b}{2a}=-\frac{-2}{2\times1}=1 \), so axis of symmetry \( x = 1 \).

1. For \( g(x)=x^{2}+6x + 8 \):

• Y - intercept: when \( x = 0 \), \( y=0 + 0+8 = 8 \), so y - intercept \( (0,8) \).
• X - intercept: solve \( x^{2}+6x + 8 = 0\), \( (x + 2)(x + 4)=0\), \( x=-2 \) or \( x=-4 \). So x - intercept \( (-2,0) \) matches.

1. For \( h(x)=-x^{2}+2x \):

• Rewrite as \( h(x)=-(x^{2}-2x)=-(x^{2}-2x + 1-1)=-(x - 1)^{2}+1 \). Axis of symmetry \( x = 1 \), maximum value 1, but the given minimum value is \( - 1 \). Wait, maybe \( h(x)=x^{2}-2x - 1 \)? No, the user's \( h(x)=-x^{2}+2x \).

Wait, let's use the given attributes:

• Axis of symmetry \( x = 1 \): for a quadratic \( ax^{2}+bx + c \), axis of symmetry is \( x=-\frac{b}{2a} \). For \( f(x)=x^{2}-2x - 8 \), \( a = 1\), \( b=-2 \), so \( x=-\frac{-2}{2\times1}=1 \). For \( h(x)=-x^{2}+2x \), \( a=-1\), \( b = 2 \), \( x=-\frac{2}{2\times(-1)} = 1 \). So axis of symmetry \( x = 1 \) applies to \( f(x) \) and \( h(x) \).

• Minimum value \( - 1 \): for a quadratic with \( a>0 \), it has a minimum. Let's check \( h(x)=-x^{2}+2x \) has \( a=-1<0 \), maximum. \( f(x)=x^{2}-2x - 8=(x - 1)^{2}-9 \), minimum - 9. \( g(x)=x^{2}+6x + 8=(x + 3)^{2}-1 \), so minimum value is \( - 1 \) (since \( a = 1>0 \), vertex at \( x=-3 \), \( g(-3)=9-18 + 8=-1 \)). Ah! So \( g(x)=x^{2}+6x + 8 \) has minimum value \( - 1 \), axis of symmetry \( x=-3 \)? Wait, no, \( x=-\frac{6}{2\times1}=-3 \). Wait, the given axis of symmetry is \( x = 1 \). So maybe the function \( h(x)=-x^{2}+2x \), let's check \( h(x)=-x^{2}+2x=-(x^{2}-2x)=-(x - 1)^{2}+1 \), axis of symmetry \( x = 1 \), maximum value 1.

• X - intercept \( (2,0) \): check \( f(x)=x^{2}-2x - 8 \) at \( x = 2 \), \( y=4-4 - 8=-8

eq0 \). \( g(x)=x^{2}+6x + 8 \) at \( x = 2 \), \( y=4 + 12+8=24
eq0 \). \( h(x)=-x^{2}+2x \) at \( x = 2 \), \( y=-4 + 4=0 \). So \( h(x) \) has x - intercept \( (2,0) \).

• X - intercept \( (-2,0) \): check \( f(x)=x^{2}-2x - 8 \) at \( x=-2 \), \( y=4 + 4-8=0 \). So \( f(x) \) has x - intercept \( (-2,0) \)? Wait, \( f(-2)=4 + 4-8=0 \), yes! So \( f(x) \) has x - intercept \( (-2,0) \)? But the option has \( (-2,0) \) and \( (2,0) \). \( h(x) \) has \( (2,0) \) (since \( h(2)=-4 + 4=0 \)).

• Y - intercept \( (0,8) \): \( g(x) \) at \( x = 0 \), \( y=8 \), so \( g(x) \) has y - intercept \( (0,8) \).

• Y - intercept \( (0,-8) \): \( f(x) \) at \( x = 0 \), \( y=-8 \), so \( f(x) \) has y - intercept \( (0,-8) \).

• Minimum value \( - 1 \): \( g(x)=x^{2}+6x + 8=(x + 3)^{2}-1 \), so minimum value \( - 1 \).

• Axis of symmetry \( x = 1 \): \( f(x)=x^{2}-2x - 8 \) (axis \( x = 1 \)) and \( h(x)=-x^{2}+2x \) (axis \( x = 1 \)).

Let's make the correct matches:

• \( f(x)=x^{2}-2x - 8 \):
• Y - intercept: \( (0,-8) \)
• X - intercept: \( (-2,0) \) (since \( f(-2)=4 + 4-8=0 \))
• Axis of symmetry: \( x = 1 \) (since \( x=-\frac{-2}{2\times1}=1 \))

• \( g(x)=x^{2}+6x + 8 \):
• Y - intercept: \( (0,8) \) (since \( g(0)=8 \))
• X - intercept: \( (-2,0) \) (since \( g(-2)=4-12 + 8=0 \))
• Minimum value: \( - 1 \) (since \( g(x)=(x + 3)^{2}-1 \))

• \( h(x)=-x^{2}+2x \):
• X - intercept: \( (2,0) \) (since \( h(2)=-4 + 4=0 \))
• Axis of symmetry: \( x = 1 \) (since \( x=-\frac{2}{2\times(-1)}=1 \))

So the matches are:

• \( f(x)=x^{2}-2x - 8 \): y - intercept \( (0,-8) \), x - intercept \( (-2,0) \), axis of symmetry \( x = 1 \)
• \( g(x)=x^{2}+6x + 8 \): y - intercept \( (0,8) \), x - intercept \( (-2,0) \), minimum value \( - 1 \)
• \( h(x)=-x^{2}+2x \): x - intercept \( (2,0) \), axis of symmetry \( x = 1 \)

(Note: There seems to be a discrepancy in the minimum value for \( f(x) \) as calculated, but we follow the given attribute options.)

Final Matches (in table form):

Function	Axis of Symmetry	Minimum Value	y - intercept	x - intercept
\( g(x)=x^{2}+6x + 8 \)	-	\( - 1 \)	\( (0,8) \)	\( (-2,0) \)
\( h(x)=-x^{2}+2x \)	\( x = 1 \)	-	-	\( (2,0) \)

(The above table shows the correct matching of each function with the given attributes based on the properties of quadratic functions: y - intercept (x = 0), x - intercept (y = 0), axis of symmetry \( x=-\frac{b}{2a} \), and vertex form for minimum/maximum value.)