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1. Recognizing function families

Warm-up

Three quick questions from earlier lessons.

Discover

f(x) = x² and g(x) = 2^x: which one has the variable in the exponent?

Key idea

Recognize the family from the formula, the table or the graph.

Polynomial: sums of terms ax^n (x in the base, whole-number powers). Rational: a quotient of polynomials (x in a denominator; vertical asymptotes). Exponential: x in the exponent (equal ratios in a table). Logarithmic: log of x (slow growth, vertical asymptote). Trigonometric: sine or cosine of x (repeats periodically).

Worked example

Name the families of 3x³ − x, 5/(x − 2), 4 · (0.5)^x, ln x and sin 2x.

polynomial, rational, exponential, logarithmic, trigonometric

Formulas.

Your turn, guided

Your turn: a table goes 1, 3, 9, 27. By what factor does it grow each step?

3

3 ÷ 1, 9 ÷ 3.

Practice ★

What kind of function is f(x) = 9 · 3^x?
1. Exponential
2. Polynomial
3. Linear
What kind of function is f(x) = 4 · 5^x?
1. Exponential
2. Polynomial
3. Linear

What kind of function is g(x) = (x + 4) / (x − 2)?
1. Rational, with a vertical asymptote at x = 2
2. Polynomial of degree 1
3. Logarithmic

Watch out!

Practice ★★

A function has f(0) = 3, f(1) = 6, f(2) = 12, f(3) = 24. Which family does it most likely belong to?
1. Exponential: the values multiply by 2 each step
2. Quadratic: the values grow faster and faster
3. Linear
A function has f(0) = 4, f(1) = 8, f(2) = 16, f(3) = 32. Which family does it most likely belong to?
1. Exponential: the values multiply by 2 each step
2. Quadratic: the values grow faster and faster
3. Linear

Boss problem. Two models predict the number of app users (in thousands) after t months: A(t) = 37t + 4 and B(t) = 4 · 2^t.

Exit ticket

Three questions to check you have got it.

What kind of function is f(x) = 9 · 3^x?
1. Exponential
2. Polynomial
3. Linear
What kind of function is f(x) = 4 · 5^x?
1. Exponential
2. Polynomial
3. Linear
What kind of function is f(x) = 9 · 5^x?
1. Exponential
2. Polynomial
3. Linear

Summary card

Function families

Base or exponent, denominator, log, periodic: each family has its sign.

Watch out: x in the exponent means exponential, not polynomial.

2. Average rate of change; where a function increases or decreases

Warm-up

Three quick questions from earlier lessons.

Discover

A car drives 120 miles in 2 hours. What is its average speed?

60 miles per hour
$\tfrac{1}{60}$ hour per mile
120 miles per hour

Key idea

The average rate of change tells how fast the output changes, per unit of input.

The average rate of change of

f

from

x = a

x = b

\dfrac{f(b) - f(a)}{b - a}

, the slope of the line through the two points of the graph.

$f$ is increasing on an interval if its graph goes up from left to right there, and decreasing if it goes down.

Worked example

Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 4$ .

f(4) - f(1) = 16 - 1 = 15

Change in the output.

4 - 1 = 3

Change in the input.

Rate

= \dfrac{15}{3} = 5

Divide: output change per unit of input.

Your turn, guided

Your turn: $f(x) = 2x^2$ , from $x = 0$ to $x = 3$ .

f(3) - f(0) = 18

f(3) = 18, f(0) = 0.

Change in

x

3

3 − 0.

Rate

= 6

18 ÷ 3.

Practice ★

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 4$ .

Watch out!

Practice ★★

A tank holds 16 liters at time 0 and $61$ liters after 5 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 29 liters at time 0 and $49$ liters after 4 minutes. What is the average rate of change of the volume, in liters per minute?
A tank holds 12 liters at time 0 and $36$ liters after 6 minutes. What is the average rate of change of the volume, in liters per minute?

$f(x) = -(x - 5)^2 + 8$ . On which interval is $f$ increasing?
1. $(-\infty, 5)$
2. $(5, \infty)$
3. $(-\infty, 8)$
$f(x) = -(x - 1)^2 + 7$ . On which interval is $f$ increasing?
1. $(-\infty, 1)$
2. $(1, \infty)$
3. $(-\infty, 7)$
$f(x) = -(x - 3)^2 + 2$ . On which interval is $f$ increasing?
1. $(-\infty, 3)$
2. $(3, \infty)$
3. $(-\infty, 2)$

Exit ticket

Three questions to check you have got it.

Find the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 6$ .
Find the average rate of change of $f(x) = 4x^2$ from $x = 1$ to $x = 5$ .
Find the average rate of change of $f(x) = 1x^2$ from $x = 1$ to $x = 6$ .

Summary card

Average rate of change

From $a$ to $b$ : $\dfrac{f(b) - f(a)}{b - a}$ , output change over input change.
Increasing: the graph rises from left to right.
Give intervals of increase with $x$ -values.

Watch out: change in $f(x)$ on top, change in $x$ below.

3. Domain and range in interval notation

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers can you put into $f(x) = \sqrt{x}$ ?

0 and the positive numbers
Every number
Only the positive numbers, not 0

Key idea

The domain is the set of inputs a function accepts. The range is the set of outputs it gives.

Leave out of the domain every

x

that makes a denominator

0

or puts a negative number under a square root. A square bracket includes the end:

[2, \infty)

contains 2; a parenthesis leaves it out:

(2, \infty)

does not.

Worked example

Find the domain of $f(x) = \sqrt{x + 2}$ .

x + 2 \ge 0

The inside of a square root must not be negative.

x \ge -2

Subtract 2 from both sides.

Domain:

[-2, \infty)

−2 is allowed (√0 = 0), so use a square bracket.

Your turn, guided

Your turn: $f(x) = \sqrt{x - 5}$ . Fill in the missing line.

x - 5 \ge 0

The inside must not be negative.

The domain starts at

x = 5

[5, \infty)

Add 5 to both sides: the domain starts at this number.

Practice ★

What is the domain of $f(x) = \sqrt{x - 9}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 5}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 6}$ ? Give it as an interval.
What is the domain of $f(x) = \sqrt{x - 4}$ ? Give it as an interval.

Watch out!

Practice ★★

What is the domain of $f(x) = \dfrac{ 9 }{x - 6}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 6 }{x - 5}$ ? Give it as an interval or a union of intervals.
What is the domain of $f(x) = \dfrac{ 4 }{x - 3}$ ? Give it as an interval or a union of intervals.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Exit ticket

Three questions to check you have got it.

What is the range of $f(x) = (x - 5)^2 + 9$ ? Give it as an interval.
What is the range of $f(x) = (x - 3)^2 + 6$ ? Give it as an interval.
What is the range of $f(x) = (x - 5)^2 + 4$ ? Give it as an interval.

Summary card

Domain and range

Domain: the inputs. Leave out zero denominators and negatives under a square root.
Range: the outputs.
$[a, \infty)$ includes $a$ ; $(a, \infty)$ does not.

Watch out: the range uses $y$ -values, not the $x$ of a vertex.

4. Vertex form of a quadratic

Warm-up

Three quick questions from earlier lessons.

Discover

$y = (x - 4)^2 + 1$ . A square is never negative. What is the smallest value of $y$ , and for which $x$ ?

$y = 1$ at $x = 4$
$y = 1$ at $x = -4$

Key idea

Vertex form shows the vertex.

In vertex form

y = a(x - h)^2 + k

the vertex is

(h, k)

. Compared with

y = ax^2

, the graph is shifted

h

right and

k

up. Watch the sign:

(x + 3)^2

means

h = -3

Worked example

$y = 2(x + 3)^2 - 5$ .

h = -3

x + 3 = x − (−3).

k = -5

Constant.

Vertex

(-3, -5)

, a minimum

a = 2 > 0.

Your turn, guided

Your turn: the $y$ -coordinate of the vertex of $y = -(x - 2)^2 + 9$ .

9

k = 9.

Practice ★

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?

What is the smallest value of $y = (x - 9)^2 - 3$ ?

Watch out!

Practice ★★

Write a quadratic with vertex $(8, 1)$ and leading coefficient 2, in vertex form.
Write a quadratic with vertex $(4, 1)$ and leading coefficient 4, in vertex form.

Exit ticket

Three questions to check you have got it.

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?
What is the $x$ -coordinate of the vertex of $y = 8(x + 5)^2 + 9$ ?

Summary card

Vertex form

$y = a(x - h)^2 + k$ : vertex $(h, k)$ .

Watch out: $(x + h)$ means the vertex is at $-h$ .

5. Linear or exponential: equal differences or equal factors

Warm-up

Three quick questions from earlier lessons.

Discover

Offer A: 1,000 dollars a day for 30 days. Offer B: 1 cent the first day, then double every day. Which pays more on day 30?

Offer B
Offer A

Key idea

Linear adds, exponential multiplies.

Over equal steps in

x

, a linear function changes by equal differences (add

m

); an exponential function changes by equal factors (multiply by

b

). A growing exponential eventually overtakes any linear function.

Worked example

$y$ : 3, 6, 12, 24.

Not linear

Differences 3, 6, 12: not equal.

Exponential, factor 2

Ratios 2, 2, 2.

24 \cdot 2 = 48

Next value.

Your turn, guided

Your turn: an exponential function has $g(0) = 5$ , $g(1) = 15$ . Find $g(2)$ .

45

Factor 3.

Practice ★

A table gives $x = 0, 1, 2, 3$ and $y = 4, 12, 36, 108$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 8 at each step
3. Neither
A table gives $x = 0, 1, 2, 3$ and $y = 8, 24, 72, 216$ . What kind of relationship is it?
1. Exponential: $y$ is multiplied by 3 at each step
2. Linear: $y$ goes up by 16 at each step
3. Neither

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .

Watch out!

Practice ★★

$f(x) = 57x + 380$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal
$f(x) = 73x + 397$ and $g(x) = 1.04^x$ . For very large $x$ , which is greater?
1. $g$ : exponential growth eventually overtakes linear growth
2. $f$ : it is greater now and grows faster
3. They stay about equal

Exit ticket

Three questions to check you have got it.

An exponential function has $g(0) = 9$ and $g(1) = 18$ . Find $g(3)$ .
An exponential function has $g(0) = 4$ and $g(1) = 20$ . Find $g(3)$ .
An exponential function has $g(0) = 5$ and $g(1) = 15$ . Find $g(3)$ .

Summary card

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

6. Linear equations in two variables and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Is $(2, 4)$ a solution of $3x + y = 10$ ?

Key idea

A linear equation in two variables is a line.

The solutions of

Ax + By = C

are the points of a line. Set

y = 0

for the $x$ -intercept and

x = 0

for the $y$ -intercept. Solving for

y

gives

y = mx + b

, with slope

m = -\tfrac{A}{B}

Worked example

$2x + 5y = 20$ .

x

-intercept

10

y = 0.

y

-intercept

4

x = 0.

Slope

-\tfrac{2}{5}

y = −(2/5)x + 4.

Your turn, guided

Your turn: the $y$ -intercept of $3x + 4y = 24$ .

6

x = 0: 4y = 24.

Practice ★

The point $(2, y)$ is a solution of $5x + y = 15$ . Find $y$ .
The point $(1, y)$ is a solution of $5x + y = 17$ . Find $y$ .

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).

Watch out!

Practice ★★

What is the slope of the line $8x + 9y = 17$ ?
What is the slope of the line $1x + 4y = 4$ ?

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $3x + 1y = 3$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $4x + 1y = 12$ (the value of $x$ where the line crosses the $x$ -axis).

Summary card

Lines from equations

Intercepts: set the other variable to 0. Slope: solve for $y$ .

Watch out: slope of $Ax + By = C$ is $-A/B$ .

7. Systems with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

$y = 2x + 1$ and $y = 2x + 5$ . Can both be true at once?

No, never
Yes, always

Key idea

Two lines meet once, never, or everywhere.

Write both equations as

y = mx + b

. Different slopes: one solution. Same slope, different intercepts: parallel lines, no solution. Same slope and intercept: the same line, infinitely many solutions.

Worked example

$y = 3x - 2$ and $6x - 2y = 4$ .

y = 3x - 2

Solve the second for y.

Infinitely many solutions

Identical.

Your turn, guided

Your turn: how many solutions has $y = 4x + 1$ , $y = -x + 6$ ? Type the number.

1

Different slopes.

Practice ★

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many

How many solutions has the system $y = 3x + 2$ and $y = -3x + 9$ ?
1. Exactly one
2. No solution
3. Infinitely many

Watch out!

Practice ★★

How many solutions has the system $y = 6x + 4$ and $12x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one
How many solutions has the system $y = 5x + 4$ and $10x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one

Exit ticket

Three questions to check you have got it.

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 4x + 8$ and $y = 4x + 10$ ?
1. No solution
2. Exactly one
3. Infinitely many

Summary card

How many solutions?

Different slopes: one. Parallel: none. Same line: infinitely many.

Watch out: same slope alone is not enough; check the intercepts.

8. Linear inequalities and systems of inequalities in two variables

Warm-up

Three quick questions from earlier lessons.

Discover

Is the point $(0, 0)$ a solution of $y > 2x + 3$ ?

Key idea

An inequality in two variables is a half-plane.

Graph

y < mx + b

y > mx + b

with a dashed boundary line, and

\le

\ge

with a solid line. Shade above for

y >

and below for

y <

(or test a point). The solutions of a system of inequalities are where the shadings overlap.

Worked example

$y \ge -x + 4$ .

Solid line

y = -x + 4

≥ includes the line.

Shade the side without the origin: above

Test (0, 0): 0 ≥ 4 is false.

Your turn, guided

Your turn: is $(1, 6)$ a solution of $y > 3x + 2$ ? Type 1 for yes, 0 for no.

1

6 > 5: yes.

Practice ★

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$

How do you graph $y \le 6x + 2$ ?
1. Solid line, shade below
2. Dashed line, shade below
3. Solid line, shade above

Watch out!

Practice ★★

Adult tickets cost 15 dollars and child tickets 5 dollars. A group can spend at most 168 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $15x + 5y \le 168$
2. $15x + 5y \ge 168$
3. $15x + 5y < 168$
Adult tickets cost 9 dollars and child tickets 3 dollars. A group can spend at most 126 dollars on $x$ adult and $y$ child tickets. Which inequality models this?
1. $9x + 3y \le 126$
2. $9x + 3y \ge 126$
3. $9x + 3y < 126$

Exit ticket

Three questions to check you have got it.

Which point is a solution of $y > 3x + 3$ ?
1. $(0, 4)$
2. $(0, 3)$
3. $(0, 2)$
Which point is a solution of $y > 5x + 7$ ?
1. $(0, 8)$
2. $(0, 7)$
3. $(0, 6)$
Which point is a solution of $y > 3x + 8$ ?
1. $(0, 9)$
2. $(0, 8)$
3. $(0, 7)$

Summary card

Inequalities in two variables

Dashed for $<, >$ ; solid for $\le, \ge$ . Test a point to choose the side.

Watch out: “at most” is $\le$ , “at least” is $\ge$ .

Answers: Functions and modeling

1. Recognizing function families

Exponential
Exponential

Rational, with a vertical asymptote at x = 2

Exponential: the values multiply by 2 each step
Exponential: the values multiply by 2 each step

$30$

Exponential
Exponential
Exponential

2. Average rate of change; where a function increases or decreases

$18$
$24$
$7$
$5$

$(-\infty, 5)$
$(-\infty, 1)$
$(-\infty, 3)$

$18$
$24$
$7$

3. Domain and range in interval notation

$[9, \infty)$
$[5, \infty)$
$[6, \infty)$
$[4, \infty)$

$(-\infty, 6) \cup (6, \infty)$
$(-\infty, 5) \cup (5, \infty)$
$(-\infty, 3) \cup (3, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

$[9, \infty)$
$[6, \infty)$
$[4, \infty)$

4. Vertex form of a quadratic

$-1$
$-9$

$-3$

$2 \left(x - 8\right)^{2} + 1$
$4 \left(x - 4\right)^{2} + 1$

$-1$
$-9$
$-5$

5. Linear or exponential: equal differences or equal factors

Exponential: $y$ is multiplied by 3 at each step
Exponential: $y$ is multiplied by 3 at each step

$72$

$g$ : exponential growth eventually overtakes linear growth
$g$ : exponential growth eventually overtakes linear growth

$72$
$500$
$135$

6. Linear equations in two variables and their graphs

$5$
$12$

$18$

$- \frac{8}{9}$
$- \frac{1}{4}$

$18$
$1$
$3$

7. Systems with one, no or infinitely many solutions

No solution
No solution

Exactly one

Infinitely many
Infinitely many

No solution
No solution
No solution

8. Linear inequalities and systems of inequalities in two variables

$(0, 4)$
$(0, 8)$

Solid line, shade below

$15x + 5y \le 168$
$9x + 3y \le 126$

$(0, 4)$
$(0, 8)$
$(0, 9)$