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Introduction to Algebra / LEVEL 4 · DIFFICULTY 4/5

Algebraic Models and Sequences

Simplify ratios and recognize additive and multiplicative growth.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST ALG 4.1Rational Expressions and RestrictionsRead the lesson
MINI QUEST ALG 4.2Exponential Growth and LogarithmsRead the lesson
MINI QUEST ALG 4.3Arithmetic Sequences and SumsRead the lesson
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STAGE ALG 4.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Rational Expressions and Restrictions

Useful preparation: Linear Inequalities

Goal: Simplify a factored ratio while preserving its domain.

Before you begin: Linear Inequalities

Understand the idea

A rational expression is a ratio of polynomials. It is undefined wherever its original denominator is zero. Canceling a common nonzero factor simplifies the rule without restoring those excluded inputs.

\displaystyle \frac{x^{2} - h^{2}}{x - h} = x + h

, only for

\displaystyle x \ne h

Choose and carry out a method

Factor first. Cancel common factors, not pieces of sums. Keep a written list of excluded inputs before using the simplified rule.

Check the reasoning

Verify that the evaluation input is allowed. A missing point can remain even when the simplified expression is a polynomial.

WORKED EXAMPLE 1

Evaluate $\displaystyle \frac{x^{2} - 9}{x - 3}$ at $\displaystyle x = 8$ .

Factor the numerator before canceling a common factor.
$\displaystyle \left(x^{2} - 9\right) = \left(x - 3\right)\, \left(x + 3\right)$ ; $\displaystyle x = 8$ is allowed.
The quotient is $\displaystyle x + 3$ , giving 11. The original restriction remains $\displaystyle x \ne 3$ .

WORKED EXAMPLE 2

Evaluate $\displaystyle \frac{x^{2} - 16}{x - 4}$ at $\displaystyle x = 9$ .

Factor the numerator before canceling a common factor.
$\displaystyle \left(x^{2} - 16\right) = \left(x - 4\right)\, \left(x + 4\right)$ ; $\displaystyle x = 9$ is allowed.
The quotient is $\displaystyle x + 4$ , giving 13. The original restriction remains $\displaystyle x \ne 4$ .

Common pitfalls

Possible mix-up: Terms can be canceled across addition.

Only whole multiplicative factors can be canceled.

Possible mix-up: Cancellation makes every input allowed.

Restrictions from the original denominator still apply.

Explain it to yourself

How can two expressions agree at all allowed inputs but have different domains?

Preview the eight practice prompts

Problem 1
Evaluate $\displaystyle \frac{x^{2} - 36}{x - 6}$ at $\displaystyle x = 11$ .
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Problem 2
Evaluate $\displaystyle \frac{x^{2} - 49}{x - 7}$ at $\displaystyle x = 12$ .
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Problem 3
Evaluate $\displaystyle \frac{x^{2} - 64}{x - 8}$ at $\displaystyle x = 13$ .
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Problem 4
Evaluate $\displaystyle \frac{x^{2} - 81}{x - 9}$ at $\displaystyle x = 14$ .
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Problem 5
Evaluate $\displaystyle \frac{x^{2} - 100}{x - 10}$ at $\displaystyle x = 15$ .
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Problem 6
Evaluate $\displaystyle \frac{x^{2} - 121}{x - 11}$ at $\displaystyle x = 16$ .
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Problem 7New context
A model gives average output $\displaystyle \frac{n^{2} - 144}{n - 12}$ for $\displaystyle n \ne 12$ . Evaluate it at the production setting $\displaystyle n = 17$ .
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Problem 8New context
A model gives average output $\displaystyle \frac{n^{2} - 169}{n - 13}$ for $\displaystyle n \ne 13$ . Evaluate it at the production setting $\displaystyle n = 18$ .
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Open stage ALG 4.1 in the student workspace →

STAGE ALG 4.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Exponential Growth and Logarithms

Useful preparation: Rational Expressions and Restrictions

Goal: Recover the number of repeated growth steps.

Before you begin: Rational Expressions and Restrictions

PARENT EQUATION

Exponential family: $\displaystyle y = b^{x}\, \left(b > 0, b \ne 1\right)$

The unshifted graph passes through (0, 1), stays positive and approaches $\displaystyle y = 0$ . It grows for $\displaystyle b > 1$ and decays for $\displaystyle 0 < b < 1$ .

How does the question’s equation change this graph?

In $\displaystyle y = a \cdot b^{x - h} + k$ , h shifts horizontally, a scales or reflects vertically, and k moves the horizontal asymptote to $\displaystyle y = k$ .

Check: Keep the base b from the question. Exponential growth multiplies by a factor; it does not add the same amount each step.

First identify this family. Then read the coefficients and shifts in your actual equation; this reference does not replace the question’s given values.

Understand the idea

Equal multiplicative changes produce an exponential model. The coefficient is the initial amount and the exponent counts repeated steps. A logarithm asks which exponent produces a given positive value.

\displaystyle A\, \left(t\right) = A_{0} \cdot b

ᵗ;

\displaystyle \operatorname{log}_{b}

(bᵗ)=t for

\displaystyle b > 0

\displaystyle b \ne 1

Choose and carry out a method

Divide by the initial amount, rewrite the resulting ratio as a power of the growth factor, and compare exponents. A logarithm generalizes this reversal when the power is less obvious.

Check the reasoning

Evaluate the model at time zero and at the proposed time. Doubling for three steps multiplies by eight, not six.

WORKED EXAMPLE 1

Solve $\displaystyle 3 \cdot 2^{x} = 48$ for x.

Isolate the exponential factor before comparing powers.
$\displaystyle 2^{x} = \frac{48}{3} = 16 = 2^{4}$ .
Therefore $\displaystyle x = 4$ . The exponent counts doublings, not the number of cells.

WORKED EXAMPLE 2

Solve $\displaystyle 4 \cdot 2^{x} = 128$ for x.

Isolate the exponential factor before comparing powers.
$\displaystyle 2^{x} = \frac{128}{4} = 32 = 2^{5}$ .
Therefore $\displaystyle x = 5$ . The exponent counts doublings, not the number of cells.

Common pitfalls

Possible mix-up: Doubling means add two each step.

It means multiply the current amount by two.

Possible mix-up: The final amount equals the exponent.

The exponent counts how many multiplications occurred.

Explain it to yourself

Compare a sequence that adds three each step with one that triples.

Preview the eight practice prompts

Problem 1
Solve $\displaystyle 6 \cdot 2^{x} = 48$ for x.
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Problem 2
Solve $\displaystyle 7 \cdot 2^{x} = 112$ for x.
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Problem 3
Solve $\displaystyle 8 \cdot 2^{x} = 256$ for x.
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Problem 4
Solve $\displaystyle 9 \cdot 2^{x} = 576$ for x.
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Problem 5
Solve $\displaystyle 10 \cdot 2^{x} = 80$ for x.
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Problem 6
Solve $\displaystyle 11 \cdot 2^{x} = 176$ for x.
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Problem 7New context
A culture starts with 12 cells and doubles every hour. After how many hours does it first have 384 cells?
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Problem 8New context
A culture starts with 13 cells and doubles every hour. After how many hours does it first have 832 cells?
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Open stage ALG 4.2 in the student workspace →

STAGE ALG 4.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Arithmetic Sequences and Sums

Useful preparation: Exponential Growth and Logarithms

Goal: Use constant differences to sum a growing arrangement.

Before you begin: Exponential Growth and Logarithms

Understand the idea

An arithmetic sequence has a fixed difference between adjacent terms. There are $\displaystyle n - 1$ gaps from the first to the nth term. Pairing terms from opposite ends produces equal pair sums.

\displaystyle a_{n} = a_{1} + \left(n - 1\right)\, d

;

\displaystyle S_{n} = \frac{n\, \left(a_{1} + a_{n}\right)}{2}

Choose and carry out a method

Find the last term, average the first and last, and multiply by the number of terms. The same pairing formula works whether the number of terms is odd or even.

Check the reasoning

Check whether the task asks for the nth term or the whole sum. For increasing positive terms, the sum lies between n times the first and n times the last.

WORKED EXAMPLE 1

Find the sum of the first 5 terms of an arithmetic sequence starting 3, 6, 9, … .

Pair the first and last terms; each pair has the same sum.
Last term= $\displaystyle 3 + \left(5 - 1\right) \cdot 3 = 15$ . Sum= $\displaystyle \frac{5\, \left(3 + 15\right)}{2}$ .
The total is 45; there are 4 increases between 5 terms.

WORKED EXAMPLE 2

Find the sum of the first 6 terms of an arithmetic sequence starting 4, 7, 10, … .

Pair the first and last terms; each pair has the same sum.
Last term= $\displaystyle 4 + \left(6 - 1\right) \cdot 3 = 19$ . Sum= $\displaystyle \frac{6\, \left(4 + 19\right)}{2}$ .
The total is 69; there are 5 increases between 6 terms.

Common pitfalls

Possible mix-up: There are n differences before term n.

There are $\displaystyle n - 1$ differences starting from term one.

Possible mix-up: The final term is the total sum.

Add all terms, for example using equal end-pairs.

Explain it to yourself

Why does the average of the endpoints equal the average of all terms?

Preview the eight practice prompts

Problem 1
Find the sum of the first 8 terms of an arithmetic sequence starting 6, 9, 12, … .
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Problem 2
Find the sum of the first 9 terms of an arithmetic sequence starting 7, 10, 13, … .
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Problem 3
Find the sum of the first 10 terms of an arithmetic sequence starting 8, 11, 14, … .
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Problem 4
Find the sum of the first 11 terms of an arithmetic sequence starting 9, 12, 15, … .
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Problem 5
Find the sum of the first 12 terms of an arithmetic sequence starting 10, 13, 16, … .
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Problem 6
Find the sum of the first 13 terms of an arithmetic sequence starting 11, 14, 17, … .
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Problem 7New context
A stepped display has 14 rows. The first row has 12 lights and every next row has 3 more lights. How many lights are used altogether?
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Problem 8New context
A stepped display has 15 rows. The first row has 13 lights and every next row has 3 more lights. How many lights are used altogether?
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Open stage ALG 4.3 in the student workspace →

Browse another level →

Algebraic Models and Sequences

Rational Expressions and Restrictions

Understand the idea

Choose and carry out a method

Check the reasoning

Evaluate x2−9x−3\displaystyle \frac{x^{2} - 9}{x - 3}x−3x2−9​ at x=8\displaystyle x = 8x=8.

Evaluate x2−16x−4\displaystyle \frac{x^{2} - 16}{x - 4}x−4x2−16​ at x=9\displaystyle x = 9x=9.

Common pitfalls

Explain it to yourself

Exponential Growth and Logarithms

Understand the idea

Choose and carry out a method

Check the reasoning

Solve 3⋅2x=48\displaystyle 3 \cdot 2^{x} = 483⋅2x=48 for x.

Solve 4⋅2x=128\displaystyle 4 \cdot 2^{x} = 1284⋅2x=128 for x.

Common pitfalls

Explain it to yourself

Arithmetic Sequences and Sums

Understand the idea

Choose and carry out a method

Check the reasoning

Find the sum of the first 5 terms of an arithmetic sequence starting 3, 6, 9, … .

Find the sum of the first 6 terms of an arithmetic sequence starting 4, 7, 10, … .

Common pitfalls

Explain it to yourself

Evaluate $\displaystyle \frac{x^{2} - 9}{x - 3}$ at $\displaystyle x = 8$ .

Evaluate $\displaystyle \frac{x^{2} - 16}{x - 4}$ at $\displaystyle x = 9$ .

Solve $\displaystyle 3 \cdot 2^{x} = 48$ for x.

Solve $\displaystyle 4 \cdot 2^{x} = 128$ for x.