Grade by Grade
C. Polynomial and Rational Functions · Rational Functions and Equations

Rational Functions and Equations: Unit Review

C2.1 C2.2 C2.3 C3.5 C3.6 C3.7 C4.1 C4.2About 40 minutes
Learning goals
  • I can analyse and sketch reciprocals of linear and quadratic functions and functions of the form (ax + b)/(cx + d).
  • I can solve rational equations graphically and algebraically, rejecting extraneous roots, and use them in applications.
  • I can solve rational inequalities with graphs and sign charts and write the solution correctly.

Words to know

vertical asymptote
x = a where the denominator is 0 (after cancelling); the graph goes to ±∞ there.
horizontal asymptote
y = b that the graph approaches as x → ±∞.
extraneous root
A value from the cleared equation that fails in the original equation.
critical value
A zero of the numerator or denominator; the sign can change only there.

Let's review!

In this unit you took the reciprocals of linear and quadratic functions and read every feature of the graph from the original: zeros became vertical asymptotes, values of ±1 stayed put, and the vertex became a turning point. You analysed functions of the form (ax + b)/(cx + d), rewrote them as transformations of y = 1/x, and built a five-step strategy for sketching any simple rational function. Then you connected roots to x-intercepts, solved rational equations algebraically (watching for extraneous roots), used them to model work, travel, cost, concentration, light and growth, and solved rational inequalities with graphs and sign charts.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For graph questions, make a quick sketch with the asymptotes dashed; for equations, write the restrictions first; for inequalities, draw a sign chart and test one value in each interval. Give exact answers unless a decimal is asked for. For each question you miss, write down why (swapped VA and x-intercept? forgot a restriction? multiplied an inequality by an expression of unknown sign?) and go back to that lesson. Finish with the 10-question quiz.

Part 1: graphs of rational functionsReciprocalsof lines (1)Reciprocalsof quadratics (2, 3)(ax + b)/(cx + d)features (4, 5)Sketchingany simple one (6)
The first six lessons build up the graphs, from the simplest reciprocal to a full sketching strategy.
Part 2: equations, inequalities and modelsRoots= x-intercepts (7)Solve by algebra(8, 9)Applications(10, 13)Inequalitiessign charts (11, 12)
The last seven lessons use those graphs to solve equations and inequalities, by graph and by algebra, and to model real situations.
Just for fun

A rational function walks into a room with two dashed lines taped to the floor. 'Don't worry,' it says, 'I'll get really close, but I promise I won't cross the vertical one.'

Reciprocals of Linear Functions

1

Find the asymptotes and y-intercept of y = 1/(3x + 9).

2

Find the invariant points of y = 1/(x − 4) and its line y = x − 4.

3

What does y = 1/(2 − x) do as x → 2⁺?

Reciprocals of Quadratic Functions with Two Zeros

1

Find the vertical asymptotes and the turning point of y = 1/(x² − 4x − 5).

2

State the range of y = 1/(x² − 4x − 5).

3

Where is y = 1/(x² − 4x − 5) positive?

Reciprocals of Quadratic Functions with One Zero or No Zeros

1

How many vertical asymptotes does y = 1/(x² + 6x + 10) have? Find its maximum point.

2

State the range of y = 1/(x² + 6x + 10).

3

Describe y = 1/(x + 2)² near x = −2.

Rational Functions y = (ax + b)/(cx + d): Asymptotes and Intercepts

1

Find the asymptotes and intercepts of y = (3x − 6)/(x + 4).

2

Describe the graph of y = (x + 1)/(3x + 3).

3

Does y = (2x − 1)/(x + 1) ever equal its horizontal asymptote value 2?

Rational Functions y = (ax + b)/(cx + d): Transformations, Intervals and Equations

1

Write y = (3x + 5)/(x + 1) as q + k/(x − h). Is it increasing or decreasing?

2

Find y = (ax + b)/(x + d) with VA x = −2, HA y = 4 and x-intercept 1.

3

Find the positive and negative intervals of y = (x + 5)/(x − 1).

Sketching Simple Rational Functions: A Complete Strategy

1

Find the hole, asymptotes and y-intercept of y = (x − 2)/(x² − 5x + 6).

2

Find the asymptotes and x-intercept of y = (4x + 1)/(x² − 1).

3

State the horizontal asymptote of y = 6x/(3x − 2).

Rational Equations and x-Intercepts: Solving with Graphs

1

Find the root of (5x − 10)/(x + 3) = 0.

2

How many solutions does 1/(x² + 4) = 1/3 have?

3

Find the intersection points of y = 6/x and y = x + 1.

Solving Rational Equations Algebraically

1

Solve 5/(x − 1) = 3/(x + 1).

2

Solve x + 12/x = 7.

3

Solve x/(x − 4) = 4/(x − 4) + 3.

Rational Equations with Quadratics, Formulas and Inverses

1

Solve (x + 3)/(x − 1) = (x − 2)/(x + 2).

2

A lens has f = 15 cm and an object is 45 cm away. Find the image distance.

3

Find the inverse of f(x) = (x − 1)/(x + 2).

Applications: Work, Travel, Average Cost and Concentration

1

Pump A empties a flooded basement in 8 h and pump B in 12 h. How long together?

2

Setup costs $300 and each item costs $4. How many items make the average cost $5?

3

20 L of punch contains 5 L of juice. How much water must be added to make it 20% juice?

Rational Inequalities: Equations versus Inequalities, and Graphical Solutions

1

Solve (x − 4)/(x + 1) < 0.

2

Is x = −1 a solution of (x − 4)/(x + 1) ≤ 0?

3

Solve 1/(x − 1) > 2 using the graph.

Solving Rational Inequalities Algebraically with Sign Charts

1

Solve 3/(x + 2) ≤ 1.

2

Solve (x + 1)/(x² − 4) ≥ 0.

3

What does the careless method (multiplying 3/(x + 2) ≤ 1 by x + 2) give, and what does it miss?

Modelling with Rational Functions and Inequalities

1

For I(d) = 400/d² (d > 0), solve I ≥ 16.

2

For C(t) = 20t/(10 + t), t ≥ 0, when is C ≥ 10?

3

Find the maximum of f(t) = 6t/(t² + 4) for t > 0.

Check your understanding

Quiz

1.

What is the vertical asymptote of y = 1/(5 − x)?

2.

What is the turning point of y = 1/(x² − 2x − 3)?

3.

Which function has no vertical asymptote?

4.

What is the horizontal asymptote of y = (5x + 2)/(2x − 7)?

5.

What happens at x = 3 on the graph of y = (x − 3)/(x² − 9)?

6.

What is the root of (x + 7)/(x − 2) = 0?

7.

What is the solution of 6/(x + 1) = 2?

8.

What is the solution of 2x/(x − 5) = 10/(x − 5) + 1?

9.

One person shovels a driveway in 30 min and another in 60 min. How long do they take together?

10.

What is the solution of (x + 2)/(x − 3) > 0?

For parents and teachers

How to use the results: look at which questions your teen missed and send them back to the matching lesson.

  • Mixing up vertical asymptotes and x-intercepts, or wrong signs near an asymptote: Reciprocals of Linear Functions.
  • Wrong turning point or range for a reciprocal of a quadratic: Reciprocals of Quadratic Functions with Two Zeros and with One Zero or No Zeros.
  • Wrong horizontal asymptote (y = b/d instead of y = a/c), or missed holes: Asymptotes and Intercepts (lesson 4) and Sketching Simple Rational Functions.
  • Division errors or 'increasing on its domain': Transformations, Intervals and Equations (lesson 5).
  • Counting a hole as a root, or reading y instead of x from a graph: Rational Equations and x-Intercepts.
  • Keeping an extraneous root or skipping restrictions: Solving Rational Equations Algebraically.
  • Errors with formulas such as the lens equation, or confusing 1/f with f⁻¹: Rational Equations with Quadratics, Formulas and Inverses.
  • Adding times instead of rates, or upstream/downstream mix-ups: Applications: Work, Travel, Average Cost and Concentration.
  • Including an asymptote in an inequality's answer, bracket errors: Rational Inequalities ... Graphical Solutions.
  • Multiplying an inequality by an expression of unknown sign: Solving Rational Inequalities Algebraically with Sign Charts.
  • Answers outside the situation (negative time, tank overflow): Modelling with Rational Functions and Inequalities.