Question

4 part a give an example of a nonlinear function. you may represent the function by giving the rule in words or by writing an equation.
part b make a graph or table to represent your function.
part c use your graph or table to help you explain why your function is nonlinear.
5 math journal how can you tell if a rule represents a function?
explain how a graph, table, or equation for a rule can help you determine whether the rule is a function.
end of lesson checklist
interactive glossary find the entry for function. show an example of a linear function, a nonlinear function, and a relationship that is not a function.

Explanation:

For Question 4 (Parts A, B, C)

Step1: Choose nonlinear function

Select $y = x^2$ (squaring input $x$).

Step2: Create value table

Pick $x$ values, compute $y$:

$x$	$y = x^2$
-1	1
0	0
1	1
2	4

Step3: Explain nonlinearity

Check rate of change:
Between $x=-2$ and $x=-1$: $\frac{1-4}{-1-(-2)} = -3$
Between $x=-1$ and $x=0$: $\frac{0-1}{0-(-1)} = -1$
Between $x=0$ and $x=1$: $\frac{1-0}{1-0} = 1$
Between $x=1$ and $x=2$: $\frac{4-1}{2-1} = 3$

A rule represents a function if every input has exactly one output.

• Graph: Use the Vertical Line Test: if any vertical line drawn on the graph intersects the graph more than once, the rule is not a function. If all vertical lines intersect at most once, it is a function.
• Table: Check that no input (x-value) is repeated with different outputs (y-values). If each input has only one output, it is a function.
• Equation: For an equation in $x$ and $y$, verify that for every valid $x$-value, there is only one corresponding $y$-value. For example, $y=2x+1$ is a function (one $y$ per $x$), but $x=y^2$ is not (one $x$ can have two $y$s, like $x=4$ has $y=2$ and $y=-2$).

• Linear function: Has a constant rate of change, follows $y=mx+b$.
• Nonlinear function: Does not have a constant rate of change, often has exponents on the input.
• Non-function: A relationship where one input has multiple outputs.

Answer:

Part A

A nonlinear function: $y = x^2$ (the output is the square of the input value)

Part B

Table representing the function:

$x$	$y$
-1	1
0	0
1	1
2	4

Part C

The function $y=x^2$ is nonlinear because the rate of change between consecutive $x$-values is not constant (it changes from -3 to -1 to 1 to 3). A linear function has a constant rate of change, so this function does not meet that requirement.

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For Question 5