Back to Exercise: Interpret the equation y = mx + b as defining a linear function

Exercises: Linear and Nonlinear Functions

Work through each section in order. Show your work where indicated. For each function, remember the key test: a function is linear when its rate of change is constant.

Grade 8·22 problems·~30 min·Common Core Math - Grade 8·container·8-f-a-3

Work through problems with immediate feedback

Recall / Warm-Up

A rule takes an input, multiplies it by 4, then adds 1. Why does this rule define a function?

Because every output is a positive number

Because each input produces exactly one output

Because the graph of the rule passes through the origin point

Because the inputs are all whole numbers

For the equation $y = 2x + 3$ , what do the numbers $2$ and $3$ represent?

$2$ is the y-intercept, the starting value, and $3$ is the slope, the rate of change

$2$ is the slope (rate of change) and $3$ is the y-intercept

$2$ is an input value you plug in and $3$ is the output value that comes out

Both are inputs to the function, since they both appear on the same side of the equal sign

In the table below, $x$ increases by $1$ each row. What is the change in $y$ from one row to the next? Table: $(0, 7), (1, 10), (2, 13), (3, 16)$ .

Fluency Practice

Classify the function $y = 6x - 5$ as linear or nonlinear.

Nonlinear, because it has a negative number in it

Nonlinear, because the graph goes down

Linear, because x appears to the first power only

It cannot be classified without a table

$A coordinate plane with the five given points plotted and a straight line drawn through them$

For $y = 2x + 1$ , the points $(0,1), (1,3), (2,5), (3,7), (4,9)$ are plotted. What is the constant rate of change (change in $y$ for each increase of $1$ in $x$ )?

A table shows $(1, 5), (2, 8), (3, 11), (4, 14)$ , where $x$ increases by $1$ each row. Is the function linear or nonlinear?

Nonlinear, because the y-values keep getting larger

Linear, because the rate of change is a constant 3

A table shows $(0, 8), (1, 6), (2, 4), (3, 2)$ , where $x$ increases by $1$ each row. What is the constant rate of change?

For $y = x^2$ , complete the table of changes in $y$ . From $x=1$ to $x=2$ : $y$ goes $1 \to 4$ , change ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . From $x=2$ to $x=3$ : $y$ goes $4 \to 9$ , change ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . From $x=3$ to $x=4$ : $y$ goes $9 \to 16$ , change ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

first change:

second change:

third change:

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