Guided Practice

A linear function is often written in the form y = mx + b, or f(x) = mx + b. The variable m represents the slope, or steepness and direction, of the line, and the variable b represents the y-intercept, which is the y-value of the point where the line crosses the y-axis.

Practice entering inputs and outputs for the function on the right.

Enter different integer inputs for the function. Then enter the corresponding outputs of the function.

f(x) = –5x – 1

Input → ← Output

Input → ← Output

Input → ← Output

A linear function has a constant slope, or a constant rate of change. For example, these points probably belong to a linear function.

The constant slope is 5.

But these points do NOT belong to a linear function.

The slope between the first two points (0, 0) and (1, 1) is 1. But the slope between the next two points (1, 1) and (2, 4) is 1.5.

Decide whether the points could represent a linear function. If so, write the possible slope of the function as an integer or fraction in lowest terms. If not, write **not linear**.

Answer:

Decide whether the points could represent a linear function. If so, write the possible slope of the function as an integer or fraction in lowest terms. If not, write **not linear**.

Answer:

Decide whether the points could represent a linear function. If so, write the possible slope of the function as an integer or fraction in lowest terms. If not, write **not linear**.

Answer:

**not linear**.

Answer:

**not linear**.

Answer:

**not linear**.

Answer:

**not linear**.

Answer:

**not linear**.

Answer:

Input

Output

x

y

Input

Output

x

y

Input

Output

x

y

Input

Output

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y

Input

Output

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y

Enter an input (x). Then press Enter or → to send the input into the linear function machine. Observe the output (y). The first 4 inputs and outputs are recorded in the table, but you can enter as many inputs as you want. Try to guess the linear function. (To clear the table, press on the top of the output column.)

When the input, x, is 1, the output, y, is .

When the input (x-value) is 2, the output (y-value) is .

When the input (x-value) is 10, the output

(y-value) is .

For this function, as x increases, y . This tells us that the slope, or the rate of change, is positive.

Clear the table by pressing the top of the output column. Enter 1, then 2, then 3, then 4 as inputs. Then consider the outputs in the table.

You can think of the slope of a linear function as the constant unit rate of change, which means the constant change in the output when the change in the input is 1.

For this function, what is the change in the output when the change in the input is 1?

The slope of this function is .

Clear the table by pressing the top of the output column. Enter –1, then 0, then 1, then 2 as inputs. Then consider the outputs in the table.

The y-intercept of a linear function is the value of the function, y, when x = 0.

For this function, what is the value of y when

x = 0?

The y-intercept of this function is .

Every linear function can be written in the form

y = mx + b, where m stands for the slope, and b stands for the y-intercept.

What linear function is represented by this function machine?

Linear function: y = x +

Use the function machine to enter all kinds of inputs to check your function!

Here is a new function machine with a different linear function. Enter 1, then 2, then 3, then 4 as inputs.

As x increases, y . This tells us that the slope, or the rate of change, is negative.

For this function, what is the change in the output when the change in the input is 1?

The slope of this function is .

For this function, what is the value of y when

x = 0?

The y-intercept of this function is .

What linear function is represented by this function machine?

Linear function: y = x +

Use the function machine to enter all kinds of inputs to check your function!

Here is a new function machine with a different linear function.

As x increases, y stays the same. This tells us that the slope, or the rate of change, is equal to .

For this function, what is the change in the output when the change in the input is 1?

The slope of this function is .

For this function, what is the value of y when

x = 0?

The y-intercept of this function is .

What linear function is represented by this function machine?

Linear function: y = x +

Use the function machine to enter all kinds of inputs to check your function!

Here is a new function machine with a different linear function.

What linear function is represented by this function machine?

Linear function: y = x +

Use the function machine to enter all kinds of inputs to check your function!

Here is a new function machine with a different linear function.

What linear function is represented by this function machine?

Linear function: y = x +

Use the function machine to enter all kinds of inputs to check your function!

Starting at 10,000 feet, an airplane descends at a constant rate for 20 min. until it touches the runway. Write a linear function to represent the elevation of the plane over time, p(t).

You can think of the slope as the ratio of any change in y divided by the corresponding change in t. The plane descends 10,000 ft in 20 min, so:

The y-intercept (where the plane starts at time

t = 0) is 10,000.

So, the function is p(t) = –500t + 10,000.

In the linear function f(x) = 2x + 3, the name of

the function is , and the input variable is .

In the linear function g(c) = 2c + 3, the name of the function is , and the input variable is .

Give each value for the function f(x) = 2x + 3.

f(2) =

f(3) =

f(–5) =

Write three different ordered pairs that represent solutions to the equation y = 2x + 3.

(, )

(, )

(, )

The slope, or rate of change, of the skier function is , because the skier's elevation changes by ft every min.

The y-intercept of the skier function (the value of y when x = 0) is , because the skier starts at an elevation of ft.

Write the linear function that represents the plane's elevation, p(t), over time in minutes, t.

p(t) = t +

Use the function for the descending plane to complete each sentence.

After 2 minutes, the plane is at feet.

After 7.68 minutes, the plane is at feet.

A plane takes off from the runway and ascends at a constant rate to an elevation of 10,000 feet in 10 minutes.

Write a linear function to represent the elevation of the plane over time in minutes, p(t). Write the slope in simplest form.

p(t) = t +

Dimitri started with a balance of –$14 in his account. He made $112 over the next 10 days at a constant rate.

Write a linear function to represent the amount in his account over time in days, a(t). Write the slope as a decimal in simplest form.

a(t) = t +

A submarine descended at a constant rate from a depth of –35 meters to a depth of –150 meters in 1 minute.

Write a linear function to represent the elevation of the submarine over time **in seconds**, s(t). Write the slope in simplest form, and round the slope to the nearest hundredth.

s(t) = t +