Question

which function can be used to model the data in this table?

$x$	$f(x)$
$-1$	$0.5$
$4$	$-7$

a $f(x)=-\frac{2}{3}x + 5.5$
b $f(x)=-\frac{3}{2}x - 1$
c $f(x)=\frac{3}{2}x + 7$
d $f(x)=-\frac{5}{2}x - 4$

Explanation:

Step1: Test option A with \( x = -3 \)

Substitute \( x = -3 \) into \( f(x)=-\frac{2}{3}x + 5.5 \):
\( f(-3)=-\frac{2}{3}(-3)+5.5 = 2 + 5.5 = 7.5 \), which is not equal to \( 3.5 \). So A is incorrect.

Step2: Test option B with \( x = -3 \)

Substitute \( x = -3 \) into \( f(x)=-\frac{3}{2}x - 1 \):
\( f(-3)=-\frac{3}{2}(-3)-1=\frac{9}{2}-1 = 4.5 - 1 = 3.5 \), which matches the table. Let's check another value.

Step3: Test option B with \( x = -1 \)

Substitute \( x = -1 \) into \( f(x)=-\frac{3}{2}x - 1 \):
\( f(-1)=-\frac{3}{2}(-1)-1=\frac{3}{2}-1 = 1.5 - 1 = 0.5 \), which matches the table. Check \( x = 4 \) too.

Step4: Test option B with \( x = 4 \)

Substitute \( x = 4 \) into \( f(x)=-\frac{3}{2}x - 1 \):
\( f(4)=-\frac{3}{2}(4)-1=-6 - 1 = -7 \), which matches the table.

Step5: (Optional) Check other options briefly (for confirmation)

For option C, with \( x = -3 \), \( f(-3)=\frac{3}{2}(-3)+7=-\frac{9}{2}+7 = 2.5
eq 3.5 \).
For option D, with \( x = -3 \), \( f(-3)=-\frac{5}{2}(-3)-4=\frac{15}{2}-4 = 7.5 - 4 = 3.5 \) (matches), but with \( x = -1 \), \( f(-1)=-\frac{5}{2}(-1)-4=\frac{5}{2}-4 = 2.5 - 4 = -1.5
eq 0.5 \). So D is incorrect.

Answer:

B. \( f(x)=-\frac{3}{2}x - 1 \)