Question

learning goals from lesson 3.2
✓ i can determine the domain and range of a linear function, both continuous and discrete and graph it (standards: a.ced.2, a.rei.10, f.if.5, f.if.7a, f.le.1b) \t how i did (circle one) \t i got it! \t i’m still learning

1. \t which of the following data sets is best described by a linear function?

a. \t { (1, 0), (2, 0), (3, 2), (4, 2) } \t c. \t { (-1, -8), (0, -6), (1, -4), (2, -2) }
b. \t { (-5, -2), (-6, 2), (-7, -2), (-8, 2) } \t d. \t { (10, 5), (11, 8), (12, 12), (13, 17) }

1. \t classify the function as discrete or continuous for the given domain. then identify the range of the function.

\t \t $y = \frac{1}{2}x + 5$, domain: $x = -4, -2, 0, 2, 4$

Explanation:

Question 12

Step1: Recall linear function slope

A linear function has a constant slope. The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). We calculate the slope for each option.

Step2: Analyze Option a

Points: \((1,0),(2,0),(3,2),(4,2)\)
Slope between \((1,0)\) and \((2,0)\): \( \frac{0 - 0}{2 - 1}=0 \)
Slope between \((2,0)\) and \((3,2)\): \( \frac{2 - 0}{3 - 2}=2 \)
Slopes are not constant. Not linear.

Step3: Analyze Option b

Points: \((-5,-2),(-6,2),(-7,-2),(-8,2)\)
Slope between \((-5,-2)\) and \((-6,2)\): \( \frac{2 - (-2)}{-6 - (-5)}=\frac{4}{-1}=-4 \)
Slope between \((-6,2)\) and \((-7,-2)\): \( \frac{-2 - 2}{-7 - (-6)}=\frac{-4}{-1}=4 \)
Slopes are not constant. Not linear.

Step4: Analyze Option c

Points: \((-1,-8),(0,-6),(1,-4),(2,-2)\)
Slope between \((-1,-8)\) and \((0,-6)\): \( \frac{-6 - (-8)}{0 - (-1)}=\frac{2}{1}=2 \)
Slope between \((0,-6)\) and \((1,-4)\): \( \frac{-4 - (-6)}{1 - 0}=\frac{2}{1}=2 \)
Slope between \((1,-4)\) and \((2,-2)\): \( \frac{-2 - (-4)}{2 - 1}=\frac{2}{1}=2 \)
Slope is constant (\( m = 2 \)). Linear.

Step5: Analyze Option d

Points: \((10,5),(11,8),(12,12),(13,17)\)
Slope between \((10,5)\) and \((11,8)\): \( \frac{8 - 5}{11 - 10}=3 \)
Slope between \((11,8)\) and \((12,12)\): \( \frac{12 - 8}{12 - 11}=4 \)
Slopes are not constant. Not linear.

Step1: Classify discrete/continuous

A discrete function has a domain of distinct, separate values (like integers, specific points). The domain here is \( x=-4,-2,0,2,4 \) (distinct points), so it's discrete.

Step2: Find range by substituting \( x \)

For \( x=-4 \): \( y=\frac{1}{2}(-4)+5=-2 + 5 = 3 \)
For \( x=-2 \): \( y=\frac{1}{2}(-2)+5=-1 + 5 = 4 \)
For \( x=0 \): \( y=\frac{1}{2}(0)+5=0 + 5 = 5 \)
For \( x=2 \): \( y=\frac{1}{2}(2)+5=1 + 5 = 6 \)
For \( x=4 \): \( y=\frac{1}{2}(4)+5=2 + 5 = 7 \)

Step3: List the range

The range is the set of \( y \)-values: \( \{3,4,5,6,7\} \)

Answer:

c. \(\{(-1, -8), (0, -6), (1, -4), (2, -2)\}\)

Question 13