Question

compare functions
name
1
$y = \frac{1}{5}x + 5$

x	y
0	3
10	5
20	7

which function has the greatest rate of change?
which has the lowest initial value?
2
$y = -0.5x - 6$

x	y
-4	-4
-8	-1
-12	2

which function has the greatest rate of change?
which has the greatest initial value?
3

x	y
-2	10
2	14
6	18

$y = 0.5x + 8$
which function has the greatest intial value?
which has the greatest rate of change?

Explanation:

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Problem 1

Step1: Find rate of change (graph)

Use points $(0,2)$ and $(10,4)$:
$\text{Rate} = \frac{4-2}{10-0} = \frac{2}{10} = 0.2$

Step2: Identify rate (equation)

For $y=\frac{1}{3}x+5$, rate = $\frac{1}{3} \approx 0.333$

Step3: Find rate (table)

Use $(-10,1)$ and $(0,3)$:
$\text{Rate} = \frac{3-1}{0-(-10)} = \frac{2}{10} = 0.2$

Step4: Find initial values (x=0)

Graph: $y=2$; Equation: $y=5$; Table: $y=3$
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Problem 2

Step1: Identify rate (equation)

For $y=-0.5x-6$, rate = $-0.5$

Step2: Find rate (graph)

Use $(-6,-3)$ and $(0,-6)$:
$\text{Rate} = \frac{-6-(-3)}{0-(-6)} = \frac{-3}{6} = -0.5$

Step3: Find rate (table)

Use $(0,-7)$ and $(-4,-4)$:
$\text{Rate} = \frac{-4-(-7)}{-4-0} = \frac{3}{-4} = -0.75$

Step4: Find initial values (x=0)

Equation: $y=-6$; Graph: $y=-6$; Table: $y=-7$
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Problem 3

Step1: Find rate (table)

Use $(-6,6)$ and $(-2,10)$:
$\text{Rate} = \frac{10-6}{-2-(-6)} = \frac{4}{4} = 1$

Step2: Identify rate (equation)

For $y=0.5x+8$, rate = $0.5$

Step3: Find rate (graph)

Use $(0,8)$ and $(10,0)$:
$\text{Rate} = \frac{0-8}{10-0} = \frac{-8}{10} = -0.8$

Step4: Find initial values (x=0)

Table: Solve $y=mx+b$, $6=1(-6)+b \to b=12$; Equation: $y=8$; Graph: $y=8$
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Answer:

1.

• Greatest rate of change: The function $y=\frac{1}{3}x+5$
• Lowest initial value: The graphed function

2.

• Greatest rate of change: The function represented by the table
• Greatest initial value: The equation $y=-0.5x-6$ and the graphed function

3.

• Greatest initial value: The function represented by the table
• Greatest rate of change: The function represented by the table