Question

topic 8: working with functions and equations
lesson 2: homework
notes or additional instructions based on whole-class discussion of homework assignment:

1. for each table, determine whether the relationship is a function. circle \function\ or

ot a function.\ then, either write
the function rule, or explain why the relationship is not a function.
tables a, b, c, d with x and y values as shown
table a, b, c, d sections with \is it a function?\ (function/not a function) and space for function rule or explanation
funded by: dana center at the university of texas at austin, learning sciences research institute at the university of illinois at chicago, agile mind, inc

Explanation:

Step1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value). So we check each table for repeated x - values with different y - values.

Step2: Analyze Table a

• Check x - values: -2, 0, 2, 4, 6, 8, 10. All x - values are unique.
• Check the relationship: Let's find the pattern. For x=-2, y = 8; x = 0,y = 0; x = 2,y=-8; x = 4,y=-16. The difference in x is 2, difference in y is - 8. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 8}{0-(-2)}=\frac{-8}{2}=-4\). Let's verify with x = 2: \(y-0=-4(x - 0)\) (using point - slope form with (0,0)). So \(y=-4x\). Check x=-2: \(y=-4\times(-2)=8\) (correct). x = 2: \(y=-4\times2=-8\) (correct). So it is a function with rule \(y = - 4x\).

Step3: Analyze Table b

• Check x - values: -3, -2, -1, 0, 1, 2, 3. All x - values are unique.
• Check y - values: All y - values are 0. So for every x, y = 0. This is a constant function \(y = 0\). Since each x has exactly one y (y = 0), it is a function.

Step4: Analyze Table c

• Check x - values: -100, -25, -4, 0, -4, -25, -100.
• For x=-4, we have y = 2 and y=-2. For x=-25, we have y = 5 and y=-5. For x=-100, we have y = 10 and y=-10. Since the same x - value has different y - values, it is not a function.

Step5: Analyze Table d

• Check x - values: -3, -2, -1, 0, 1, 2, 3. All x - values are unique (even though - 3, - 2, - 1,0,1,2,3 are all present, each x has only one y - value which is 12). So for every x, y = 12. This is a constant function \(y = 12\). Since each x has exactly one y (y = 12), it is a function.

Answer:

Table a:

• Is it a function? $\boldsymbol{\text{Function}}$
• Function rule or explanation: \(y=-4x\) (each \(x\) has one \(y\), and the relationship is linear with slope - 4 and y - intercept 0)

Table b:

• Is it a function? $\boldsymbol{\text{Function}}$
• Function rule or explanation: \(y = 0\) (each \(x\) maps to 0, so each \(x\) has exactly one \(y\))

Table c:

• Is it a function? $\boldsymbol{\text{Not a function}}$
• Function rule or explanation: For \(x=-4\), \(y = 2\) and \(y=-2\); for \(x=-25\), \(y = 5\) and \(y=-5\); for \(x=-100\), \(y = 10\) and \(y=-10\). Repeated \(x\) - values have different \(y\) - values.

Table d:

• Is it a function? $\boldsymbol{\text{Function}}$
• Function rule or explanation: \(y = 12\) (each \(x\) maps to 12, so each \(x\) has exactly one \(y\))