Comparing functions

High School Math 1 · Unit 12 · comparing-function-families · Teacher edition

comparing-function-families

Name ______________________________Date ______________

1.
A table gives $x = 0, 1, 2, 3$ and $y = 2, 5, 10, 17$ . Which family does the table belong to?
1. (A)
  Quadratic
2. (B)
  Linear
3. (C)
  Exponential
4. (D)
  None of these
2.
A table gives $x = 0, 1, 2, 3$ and $y = 4, 12, 36, 108$ . The pattern continues. What is $y$ when $x = 5$ ?
Answer: ______________
3.
Let $L(x) = 20x$ and $E(x) = 2^{x}$ . For small positive $x$ , $L(x)$ is larger. What is the smallest positive integer $x$ for which $E(x) > L(x)$ ?
Answer: ______________
4.
A table gives $x = 1, 2, 3, 4$ and $y = 5, 8, 11, 15$ . Which statement about the table is correct?
1. (A)
  Linear, because the differences are all about $3$
2. (B)
  Exponential, because $y$ keeps increasing
3. (C)
  Not linear, because the first differences $3, 3, 4$ are not all equal
4. (D)
  Quadratic, because the second differences are $0$ and $1$
5.
A quadratic function $f$ satisfies $f(0) = 3$ , $f(1) = 6$ , and $f(2) = 13$ . What is $f(5)$ ?
1. (A)
  $43$
2. (B)
  $48$
3. (C)
  $53$
4. (D)
  $58$
5. (E)
  $63$

Answer key — Comparing functions

1.
(A)
Quadratic
comparing-function-families-01
First differences $3, 5, 7$ ; second differences $2, 2$ — constant. The table is quadratic ( $y = x^2 + 2x + 2$ fits). (Ratios $2.5, 2, 1.7$ are not constant, so it is not exponential.)
2.
972comparing-function-families-02
Constant ratio $3$ , so $y = 4 \cdot 3^x$ . At $x = 5$ : $4 \cdot 243 = \mathbf{972}$ (or $108 \to 324 \to 972$ ).
3.
8comparing-function-families-03
$x = 7$ : $2^7 = 128 < 140$ . $x = 8$ : $2^8 = 256 > 160$ . The smallest positive integer is $\mathbf{8}$ . From here on the exponential doubles each step while the line adds only $20$ , so it never falls behind again.
4.
(C)
Not linear, because the first differences $3, 3, 4$ are not all equal
comparing-function-families-04
The first differences $3, 3, 4$ are not all equal, so the table is not linear. (“About $3$ ” is not a rule — linear means exactly equal. Increasing $y$ alone does not make a table exponential; the ratios are not constant. Second differences $0, 1$ are not constant, so it is not quadratic either.)
5.
(D)
$58$
comparing-function-families-05
First differences $3, 7, 11, 15, 19$ (each up by $4$ ), so $f(3) = 24$ , $f(4) = 39$ , $f(5) = \mathbf{58}$ . Check with the formula $f(x) = 2x^2 + x + 3$ : $f(5) = 50 + 5 + 3 = 58$ . ✓ ( $43$ uses second difference $3$ ; $63$ adds one difference too many.)