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Part A,end emphasis,

Jamal is creating an input/output table for a linear function. Enter a number in each box to complete the table.

Question 1 Response area with 3 text input boxes
Input

Output

negative 6 negative 22
negative 4

negative 16

negative 1

5

32


Question 2
,begin emphasis,Part B,end emphasis,
Choose ,begin emphasis,all,end emphasis, linear functions with a slope that is greater than Jamal's linear function.

Question 2 Answer options with 6 options
A.
y equals 6

B.
y equals 5 x

C.
y equals negative 4 x

D.
y equals 4 x plus 3

E.
y equals x plus 10

F.
y equals 8-thirds x plus 4

1 answer

To complete Jamal's input/output table, we first need to identify the consistent pattern in the inputs and corresponding outputs to determine the linear function.

Let's examine the given values in the table:

  • For Input = -6, Output = -22
  • For Input = -4, Output = ?
  • For Input = -1, Output = ?
  • For Input = 5, Output = 32

Step 1: Find the slope (m)

  1. Between (-6, -22) and (5, 32): \[ m = \frac{(32 - (-22))}{(5 - (-6))} = \frac{32 + 22}{5 + 6} = \frac{54}{11} = \frac{54}{11} = 4.909 \text{ (approx.)} \]

So, the slope is approximately 4.909.

Step 2: Find the y-intercept (b)

Using the point-slope form of the linear equation \(y = mx + b\) and one of the points, we can solve for \(b\). Let's use the point (5, 32): \[ 32 = \frac{54}{11}(5) + b \] Calculating: \[ 32 = \frac{270}{11} + b \] To isolate \(b\): \[ b = 32 - \frac{270}{11} \] Convert 32 into a fraction: \[ 32 = \frac{352}{11} \] Thus, \[ b = \frac{352}{11} - \frac{270}{11} = \frac{82}{11} \]

Step 3: Writing the linear equation

Now we can write the equation based on the slope and y-intercept: \[ y = \frac{54}{11}x + \frac{82}{11} \]

Step 4: Compute missing outputs

  1. For Input = -4: \[ y = \frac{54}{11}(-4) + \frac{82}{11} = -\frac{216}{11} + \frac{82}{11} = -\frac{134}{11} \approx -12.18 \]

  2. For Input = -1: \[ y = \frac{54}{11}(-1) + \frac{82}{11} = -\frac{54}{11} + \frac{82}{11} = \frac{28}{11} \approx 2.55 \]

  3. For Input = 5 was already given: Output = 32

  4. The last Input of -1 is given: Output = 35.5 (or 32)

Now filling the table

  • Input = -4: Output = approximately -12.18 (or -11.73)
  • Input = -1: Output = approximately 2.55 (or 2.45)

Final table entries:

  • Input: -6, Output: -22
  • Input: -4, Output: approximately -5 (to match the question's integer output)
  • Input: -1, Output: approximately 2.5
  • Input: 5, Output: 32

Question 2: Which linear functions have a greater slope than approximately 4.909?

Answer:

B. \(y = 5x\) (Slope = 5) D. \(y = 4x + 3\) (Slope = 4)

E. \(y = x + 10\) (Slope = 1)

F. \(y = \frac{8}{3}x + 4\) (Slope = approx. 2.67)

Thus the correct selections based on their slopes would be:

  • B: \(y = 5x\)
  • D: \(y = 4x + 3\)

These two functions have slopes greater than Jamal's linear function's slope.