Question

6.3 comparing functions

1. function a is shown by the ordered pairs:

(1,5), (2,8), (3,11), (4,14)
a. write the function rule.
b. how much does the output increase for each increase of 1 in the input?
2.

x	y
2	6
4	10
6	14

a. the table represents a function.
find the rule of the function.
b. state the slope.

1. function b is given by the equation: y=2x+4. function c is shown in the table:

x	y
2	5
4	9

a. find the slope of each function.
b: m= c: m=
b. which function is increasing faster?

1. function d is shown by a graph that crosses the y - axis at 6 and increases by 3 units up for every 2 units to the right.

function e is given by: y=1.5x+2
a. write the equation for function d
b. which function has the greater rate of change?

Explanation:

Problem 1

Step1: Identify slope (rate of change)

Slope $m = \frac{8-5}{2-1} = 3$

Step2: Find y-intercept (b)

Use $(1,5)$: $5 = 3(1) + b \implies b=2$

Step3: State function rule

$y = 3x + 2$

Step4: Output increase equals slope

Output increase = slope = 3

Problem 2

Step1: Calculate slope of the table

$m = \frac{6-2}{2-0} = 2$

Step2: Find function rule

Use $(0,2)$: $y = 2x + 2$

Step3: State the slope

Slope = 2

Problem 3

Step1: Slope of Function B

From $y=2x+4$, slope $m=2$

Step2: Slope of Function C

$m = \frac{5-1}{2-0} = 2$

Step3: Compare slopes

Both slopes are equal, so neither is faster.

Problem 4

Step1: Find slope of Function D

$m = \frac{3}{2} = 1.5$; y-intercept $b=6$, so equation: $y=1.5x+6$

Step2: Compare rates of change

Slope of D = 1.5, slope of E = 1.5; rates are equal.

Answer:

1. a. $y = 3x + 2$

b. 3

1. A. $y = 2x + 2$

B. 2

1. a. B: $m=2$, C: $m=2$

b. Neither function increases faster; they have the same rate of change.

1. a. $y = 1.5x + 6$

b. Both functions have the same rate of change (1.5), so neither is greater.