QuestionTerri's function machine gives outputs: 3 → 8, 10 → 29, 20 → 59. Find outputs for 5, -1, and $x$ ; also, write the equation.

Studdy Solution

STEP 1

Assumptions1. The function machine takes an input and gives an output. . The output is dependent on the input in a consistent mathematical way.
3. The given inputs and their corresponding outputs are (3,8), (10,29), and (20,59).

STEP 2

First, we need to find the pattern or relationship between the inputs and outputs. We can start by comparing the differences between the outputs and inputs.
$Difference = Output - Input$

STEP 3

Calculate the differences for the given inputs and outputs.
For input3, output8 $Difference =8 -3 =5$ For input10, output29 $Difference =29 -10 =19$ For input20, output59 $Difference =59 -20 =39$

STEP 4

Observe the pattern in the differences. We can see that the difference increases by10 each time the input increases by7. This suggests that the function might be linear, with a slope of10/7.

STEP 5

To find the y-intercept (b) of the linear function, we can use the formula $b = Output - (lope \\times Input)$

STEP 6

Substitute the values of slope (10/), output (8), and input (3) into the formula to calculate the y-intercept.
$b =8 - ((10/) \\times3)$

STEP 7

Calculate the y-intercept.
$b = - ((10/7) \\times3) = - (30/7) =56/7 -30/7 =26/7$

STEP 8

Now that we have the slope and y-intercept, we can write the equation of the function machine. The general form of a linear equation is $y = mx + b$

STEP 9

Substitute the values of slope (m =/7) and y-intercept (b =26/7) into the equation.
$y = (/7)x +26/7$ This is the equation for Terri's machine.

STEP 10

To find the output for an input of5, substitute x =5 into the equation.
$y = (10/7) \\times5 +26/7$

STEP 11

Calculate the output for x =5.
$y = (10/7) \\times5 +26/7 =50/7 +26/7 =76/7 =10.86$ So, the output for an input of5 is approximately10.86.

STEP 12

To find the output for an input of -, substitute x = - into the equation.
$y = (10/7) \\times - +26/7$

STEP 13

Calculate the output for x = -.
$y = (10/7) \\times - +26/7 = -10/7 +26/7 =16/7 =2.29$ So, the output for an input of - is approximately2.29.

STEP 14

To find the output for an input of x, substitute x = x into the equation.
$y = (10/7)x +26/7$ So, the output for any input x is given by the equation y = (10/7)x +26/7.

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