Question

understanding linear and nonlinear functions
for problems 1–4, look at each graph and determine whether it represents a
linear or nonlinear function. explain your reasoning.
1
2
3
4
for problems 5 and 6, graph the function shown in each table to determine if the
function is linear or nonlinear. explain your reasoning.
5
input (x) \t0 \t1 \t2
output (y) \t1 \t2 \t5

Explanation:

1.

Step1: Check graph shape

The graph is a curved line (cubic curve, $y=x^3$).

Step2: Apply linear function rule

Linear functions have straight-line graphs. This curve is not straight, so it is nonlinear.

2.

Step1: Check vertical line test

A vertical line (e.g., $x=0$) intersects the graph at infinitely many points.

Step2: Apply function definition

A function has only one output per input. This fails the test, so it is not a function, and thus not a linear function.

3.

Step1: Check graph shape

The graph is a U-shaped curve (parabola, $y=x^2$).

Step2: Apply linear function rule

Linear functions have straight-line graphs. This curve is not straight, so it is nonlinear.

4.

Step1: Check graph shape

The graph is a straight line (follows $y=\frac{1}{2}x+1$).

Step2: Apply linear function rule

Linear functions are defined by straight-line graphs with constant slope, so this is linear.

5.

Step1: Calculate rate of change

Between $x=0$ and $x=1$: $\frac{2-1}{1-0}=1$
Between $x=1$ and $x=2$: $\frac{5-2}{2-1}=3$

Step2: Apply linear function rule

Linear functions have a constant rate of change. Here, the rate changes, so it is nonlinear.

Step3: Verify with graph

Plotting points $(0,1)$, $(1,2)$, $(2,5)$ gives a curved line, confirming it is nonlinear.

Answer:

1. Nonlinear function
2. Nonlinear (not a function)
3. Nonlinear function
4. Linear function
5. Nonlinear function