Question

topic: writing recursive equations for quadratic functions. identify whether the table represents a linear or quadratic function. if the function is linear, write both the explicit and recursive equations. if the function is quadratic, write only the recursive equation. 18. x f(x) 1 0 2 3 3 6 4 9 5 12 type of function: equation(s): 19. x f(x) 1 7 2 10 3 16 4 25 5 37 type of function: equation(s): 20. x f(x) 1 8 2 10 3 12 4 14 5 16 type of function: equation(s): 21. x f(x) 1 28 2 40 3 54 4 70 5 88 type of function: equation(s):

Explanation:

Step1: Check differences for 18

Find differences between consecutive $f(x)$ - values. For $x = 1$ to $x = 2$, $f(2)-f(1)=3 - 0=3$. For $x = 2$ to $x = 3$, $f(3)-f(2)=6 - 3 = 3$. Since the first - differences are constant ($3$), it's linear.

Step2: Find explicit equation for 18

The slope $m = 3$ and when $x = 1$, $y = 0$. Using the point - slope form $y - y_1=m(x - x_1)$, the explicit equation is $f(x)=3(x - 1)=3x-3$.

Step3: Find recursive equation for 18

The recursive equation for a linear function is $f(n)=f(n - 1)+m$ with $f(1)=0$ and $m = 3$, so $f(n)=f(n - 1)+3,f(1)=0$.

Step4: Check differences for 19

First - differences: $f(2)-f(1)=10 - 7 = 3$, $f(3)-f(2)=16 - 10 = 6$, $f(4)-f(3)=25 - 16 = 9$. First - differences are not constant. Second - differences: $6 - 3=3$, $9 - 6 = 3$. Since second - differences are constant, it's quadratic.

Step5: Find recursive equation for 19

Let $a_n=f(n)$. We know $a_1 = 7$. $a_2=a_1+3$, $a_3=a_2 + 6$, $a_4=a_3+9$. The pattern for the increment is $3(n - 1)$. The recursive equation is $a_n=a_{n - 1}+3(n - 1),a_1 = 7$.

Step6: Check differences for 20

Find differences: $f(2)-f(1)=10 - 8 = 2$, $f(3)-f(2)=12 - 10 = 2$. Since first - differences are constant ($2$), it's linear.

Step7: Find explicit equation for 20

The slope $m = 2$ and when $x = 1$, $y = 8$. Using point - slope form, $f(x)=2(x - 1)+8=2x + 6$.

Step8: Find recursive equation for 20

The recursive equation is $f(n)=f(n - 1)+2,f(1)=8$.

Step9: Check differences for 21

First - differences: $f(2)-f(1)=40 - 28 = 12$, $f(3)-f(2)=54 - 40 = 14$, $f(4)-f(3)=70 - 54 = 16$. First - differences are not constant. Second - differences: $14 - 12 = 2$, $16 - 14 = 2$. Since second - differences are constant, it's quadratic.

Step10: Find recursive equation for 21

Let $a_n=f(n)$. $a_1 = 28$. $a_2=a_1+12$, $a_3=a_2+14$, $a_4=a_3+16$. The increment increases by 2 each time. The recursive equation is $a_n=a_{n - 1}+10 + 2(n - 1),a_1 = 28$.

Answer:

18.
Type of function: Linear
Equation(s): Explicit: $f(x)=3x - 3$, Recursive: $f(n)=f(n - 1)+3,f(1)=0$
19.
Type of function: Quadratic
Equation(s): Recursive: $a_n=a_{n - 1}+3(n - 1),a_1 = 7$
20.
Type of function: Linear
Equation(s): Explicit: $f(x)=2x + 6$, Recursive: $f(n)=f(n - 1)+2,f(1)=8$
21.
Type of function: Quadratic
Equation(s): Recursive: $a_n=a_{n - 1}+10 + 2(n - 1),a_1 = 28$