Introduction to Linear Polynomials · Lesson 7 of 7
Chapter Summary and Practice
“Check whether your linear thinking is still moving in the right direction.”
• Recall the meaning of linear polynomials and linear patterns. • Connect linear growth, linear decay and linear relationships. • Interpret slope and y-intercept from equations and graphs. • Move confidently between verbal, tabular, algebraic and graphical forms. • Solve mixed problems involving linear polynomials and relationships.
This chapter connects several ideas that may first seem separate: algebraic expressions, degree, linear polynomials, patterns, growth and decay, equations in two variables, and straight-line graphs. The unifying idea is constant change.
Algebraic Expressions and Polynomials
An algebraic expression combines numbers, variables and operations. A univariate polynomial contains one variable, and its degree is the highest power of that variable with non-zero coefficient.
Linear Polynomial
A linear polynomial has degree 1. Equal changes in x produce equal changes in p(x).
Linear Pattern
A linear pattern is a sequence whose consecutive terms differ by a constant amount. Its nth term is a linear expression in n.
Linear Growth and Decay
If the constant change is positive, the quantity grows linearly. If it is negative, the quantity decays linearly.
Linear Relationship
The coefficient a is the constant rate of change, while b is the value of y when x = 0.
Slope and y-Intercept
| Feature | Meaning |
|---|---|
| a | Slope or constant rate of change |
| b | y-intercept |
| a > 0 | Linear growth |
| a < 0 | Linear decay |
| b = 0 | Line passes through origin |
| Same a, different b | Parallel lines |
How to Choose the Right Approach
| Problem type | What to do |
|---|---|
| Given an algebraic expression | Identify variable, terms, coefficients and degree |
| Given equal-step sequence | Check constant difference and find nth-term rule |
| Given starting value and fixed increase/decrease | Write initial value ± rate × steps |
| Given two points | Find a from change in y/change in x, then find b |
| Asked to graph y = ax + b | Choose two points, plot and join |
| Asked about graph behaviour | Interpret a as slope and b as y-intercept |
• Calling every algebraic expression a polynomial, • Forgetting that degree means the highest non-zero power, • Confusing a linear polynomial with a linear equation, • Missing the initial value in a growth or decay model, • Treating b as the slope instead of the y-intercept, • Assuming lines with different slopes can be parallel, • Forgetting that a point must satisfy the equation to lie on the graph,
Guided Practice
Problem
Classify p(x) = 7x − 4.
- 1.The highest power of x is 1.
- 2.Therefore the degree is 1.
- 3.So p(x) is a linear polynomial.
Problem
A pattern has 6, 10, 14, 18, ... objects. Find the nth-term rule.
- 1.The constant difference is 4.
- 2.A linear rule has form 4n + b.
- 3.Using n = 1 gives 6 = 4 + b, so b = 2.
- 4.Therefore the nth term is 4n + 2.
Problem
You begin with ₹800 and save ₹250 each month. How much will you have after 6 months and after 2 years?
- 1.Model: A(n) = 800 + 250n.
- 2.After 6 months: A(6) = 800 + 1500 = ₹2300.
- 3.Two years = 24 months.
- 4.A(24) = 800 + 6000 = ₹6800.
Problem
The graph of p(x) = ax + b passes through (1,5) and (3,11). Find p(x).
- 1.a = (11 − 5)/(3 − 1) = 6/2 = 3.
- 2.Use point (1,5): 5 = 3(1) + b.
- 3.So b = 2.
- 4.Therefore p(x) = 3x + 2.
Problem
For p(x) = 3x + 2, find where the graph cuts the axes.
- 1.On the y-axis, x = 0, so y = 2. The point is (0,2).
- 2.On the x-axis, y = 0, so 3x + 2 = 0.
- 3.x = −2/3.
- 4.The x-intercept is (−2/3,0).
Practice Problems
- Write a degree-3 polynomial in x whose x² coefficient is −7.
- Evaluate 5x² − 3x + 7 at x = 1.
- Evaluate 4t³ − t² + 6 at t = a.
- If multiplying a number by 5/2 and then adding 2/3 gives −7/12, find the number.
- A positive number is 5 times another. If 21 is added to both, one new number becomes twice the other. Find the original numbers.
- You begin with ₹800 and save ₹250 every month. Find the amount after 6 months and 2 years, and write the linear rule.
- A two-digit number has digits differing by 3. Reversing the digits and adding the new number to the original gives 143. Find the possible numbers.
- Draw y = −3x + 4, 2y = 4x + 7, 5y = 6x − 10 and 3y = 6x − 11. Identify slopes and y-intercepts and decide which lines are parallel.
- The Fahrenheit temperature y and Kelvin temperature x satisfy y = 9/5(x − 273) + 32. Find y when x = 313, and find x when y = 158.
- For constant force 3 units, work w and distance d satisfy w = 3d. Draw the graph and find the work done at d = 2.
- The graph of p(x) passes through (1,5) and (3,11). Find p(x), then find where it cuts both axes.
- A matchstick pattern is formed by adding one hexagon at each stage, sharing one side with the previous hexagon. Determine the number of matchsticks in Stages 1 to 5 and find a rule for Stage n.
- Let p(x) = ax + b pass through (2,3) and (6,11). Let q(x) be parallel to p(x) and pass through (4,−1). Find both polynomials and their x-intercepts.
- Investigate all functions f(x) = ax + a with a > 0. What common point do their graphs pass through?
Quiz
A polynomial of degree 1 is called:
A sequence with constant difference is called:
In y = ax + b, what does b represent?
What happens if two lines have the same slope but different y-intercepts?
A line with negative slope represents:
Check that you can identify degree, evaluate polynomials, recognise constant differences, build growth and decay models, determine y = ax + b from two points, plot linear equations, and explain slope, y-intercept and parallel lines in words.
Key Takeaways
• Degree 1 gives a linear polynomial. • Linear patterns have constant difference. • Linear growth and decay are constant-rate change in opposite directions. • A linear relationship is written y = ax + b. • a is the slope and b is the y-intercept. • The graph of a linear relationship is a straight line. • Equal slopes with different y-intercepts produce parallel lines.
This completes Introduction to Linear Polynomials. Continue by practising how the same linear idea appears as an expression, a sequence, a table, an equation and a graph.
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Visualizing Linear Relationships
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