Question

question: 2
consider the table of a function ( f(x) ).

( x )	( f(x) )
-1	1
0	0
1	1
2	4

is the function linear, quadratic, or neither?
a. linear
b. quadratic
c. neither

Explanation:

Step1: Check for linear function

A linear function has a constant slope. The slope between \(x=-2\) and \(x=-1\) is \(\frac{1 - 4}{-1 - (-2)}=\frac{-3}{1}=-3\). Between \(x=-1\) and \(x = 0\), slope is \(\frac{0 - 1}{0 - (-1)}=\frac{-1}{1}=-1\). Slopes are not constant, so not linear.

Step2: Check for quadratic function

A quadratic function has the form \(f(x)=ax^{2}+bx + c\). Let's test the points. For \(x = 0\), \(f(0)=0=a(0)^{2}+b(0)+c\), so \(c = 0\). For \(x = 1\), \(f(1)=1=a(1)^{2}+b(1)\), so \(a + b=1\). For \(x = 2\), \(f(2)=4=a(2)^{2}+b(2)\), so \(4a+2b = 4\) or \(2a + b=2\). Subtract \(a + b=1\) from \(2a + b=2\): \(a=1\), then \(b = 0\). So \(f(x)=x^{2}\), which is quadratic. The values match: \(f(-2)=(-2)^{2}=4\), \(f(-1)=(-1)^{2}=1\), etc.

Answer:

B. Quadratic