Resources For Teachers For Tutors For Students & Parents Pricing
Create Free Account
Pre-Algebra Functions (introductory)

Modelling linear relationships (rate of change and initial value)

20 practice questions 0 video lessons Theory + worked examples

Modelling Linear Relationships

Texas Pre-Algebra (TEKS) • Standard 8.4(C) • Functions

Modelling Linear Relationships is a topic in Functions in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.4(C), which requires students to model linear relationships using rate of change and initial value.

A linear model uses the rate of change as the slope and the initial value as the \(y\)-intercept.

Texas Pre-Algebra (TEKS) › Functions › Modelling Linear Relationships  —  Standard 8.4(C)

Practice 20 questions
Practice questions

Every question with a fully worked solution.

Start practising
Create a free accountTrack your progress and save your work as you go.
Create free account

Theory

A linear model \(y=mx+b\) uses the rate of change as the slope \(m\) and the initial value as the intercept \(b\).

Start amount is \(b\); per-unit change is \(m\).
Modelling a linear relationship The initial value is the y-intercept and the rate of change is the slope. hours $ start $5 $4/hour
A real-world linear model.
Rate of change & initial value Rate of change & initial value Rate of change & initial value initial value = y-intercept (b) rate of change = slope (m) y = (rate)·x + (start)
Rate and initial value.

The model:

\[y=(\text{rate})\,x+(\text{initial value})\]
y equals the rate times x plus the initial value
Units matter: the slope carries a rate.

How to build a linear model

  1. Find the starting value (the intercept).
  2. Find the constant rate of change (the slope).
  3. Write \(y=mx+b\).
  4. Substitute to make predictions.
Example 1 — Build a model
A plan costs \(\$5\) plus \(\$4\)/hour. Write the model.
Solution

Start plus rate times hours.

\(y\)\(=\)\(4x+5\)
y equals 4 x plus 5
Example 2 — Interpret slope
In \(y=4x+5\), what is \(4\)?
Solution

The rate — \(\$4\) per hour.

4 dollars per hour
Example 3 — Interpret intercept
In \(y=4x+5\), what is \(5\)?
Solution

The initial value — \(\$5\) to start.

5 dollars to start
Example 4 — Predict
Find the cost for \(3\) hours.
Solution

Substitute \(x=3\).

\(4(3)+5\)\(=\)\(\$17\)
17 dollars

Common pitfalls

The initial value is at \(x=0\).
The slope is a rate with units.
Keep start and rate in the right places.

Frequently asked questions

What is the initial value?

The \(y\)-intercept, the value at \(x=0\).

What is the rate of change?

The slope.

How do I write a linear model?

\(y=(\text{rate})x+(\text{start})\).

\(y=4x+5\): cost for 3 hours?

\(\$17\).

More in Functions (introductory)
Definition of a function
20 questions
Linear vs. non-linear functions
20 questions
Graphing linear equations (slope-intercept form)
20 questions
Interpreting graphs of functions
20 questions