Question

which type of relationship is shown in the table?
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
y | 8 | 19 | 28 | 35 | 40 | 43 | 44 | 43 | 40
select the correct answer. for help, see this worked example.

• linear function with a positive slope
• quadratic function with a positive value of a
• quadratic function with a negative value of a
• square root function

Explanation:

Step1: Check linearity (rate of change)

Calculate first differences of $y$:
$19-8=11$, $28-19=9$, $35-28=7$, $40-35=5$, $43-40=3$, $44-43=1$, $43-44=-1$, $40-43=-3$
First differences are not constant, so not linear.

Step2: Check quadratic (2nd differences)

Calculate second differences (differences of first differences):
$9-11=-2$, $7-9=-2$, $5-7=-2$, $3-5=-2$, $1-3=-2$, $-1-1=-2$, $-3-(-1)=-2$
Second differences are constant ($-2$), so it is quadratic.

Step3: Determine sign of $a$

For a quadratic $y=ax^2+bx+c$, the second difference equals $2a$.
$2a=-2$
Solve for $a$: $a=\frac{-2}{2}=-1$
$a$ is negative, and the $y$-values rise then fall (parabola opens downward).

Answer:

quadratic function with a negative value of $a$