Section 1

Quadratics.

A quadratic has three outfits.

Standard

$y=ax^2+bx+c$

c is the y-intercept.

Example $y=x^2-2x$$-3$

Factored

$y=a(x-p)(x-q)$

p and q are the x-intercepts.

Example $y=(x$$-4$$)(x$$+2$$)$

Vertex

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

(h, k) is the vertex.

Example $y=(x$$-1$$)^2$$+2$

Tip

Each form hands you a different fact for free. Match the question to the form, and convert only when you must.

The a controls the shape.

Standard

$y=$$a$$x^2+bx+c$

Factored

$y=$$a$$(x-p)(x-q)$

Vertex

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

NARROW

larger a

WIDE

smaller a

OPENS UP

a is positive

OPENS DOWN

a is negative

Which point is an x-intercept?

Problem · your turn: 30 sec, then GO

Which of the following is an x-intercept of the graph of $y=(x-4)(x+2)$?

$(x-4)(x+2)=0$ so $x=4$ or $x=-2$

A$(4,\,0)$✓

B$(2,\,0)$sign slip

C$(-4,\,0)$sign slip

D$(0,\,-8)$y-intercept

Set each factor to $0$: $x-4=0$ gives $x=4$, $x+2=0$ gives $x=-2$. The x-intercepts are $(4,0)$ and $(-2,0)$; only $(4,0)$ is listed. Note $(0,-8)$ is the y-intercept.

Or just let Desmos show you.

1

$y=(x-4)(x+2)$

Type the equation and Desmos marks the x-intercepts for you. Great for checking your work when the problem gives you numbers to type.

No numbers? Fundamentals win.

Problem · your turn: 30 sec, then GO

The graph of $y=ax^2+bx+c$ is shown. Which of the following must be true?

A$ac>0$✓

B$ac<0$

C$ac=1$

D$ac=0$needs c = 0

The graph opens down, so $a<0$. The y-intercept is $c$, and it sits below the x-axis, so $c<0$. A negative times a negative is positive, so $ac>0$. We never need the actual values.

The Discriminant.

Quadratic equation

$ax^2+bx+c=0$

→

Discriminant

$D$$=b^2-4ac$

How many solutions?

Problem · your turn: 30 sec, then GO

How many real solutions does $2x^2-5x+4=0$ have?

It does not factor nicely, so check the discriminant. Here $a=2,\ b=-5,\ c=4$.

$D$$=b^2-4ac=(-5)^2-4(2)(4)=25-32=$$-7$

A0✓

B1that needs D = 0

C2that needs D > 0

DCannot tell

$D$$=b^2-4ac=25-32=-7$. Since $D<0$, there are no real solutions. The numbers do not factor, so the discriminant is the quick way to know.

Or graph it on Desmos.

1

$y=2x^2-5x+4$

The parabola never touches the x-axis, so there are no real solutions. That is exactly what $D<0$ told us.

Which k gives no solutions?

Problem · your turn: 30 sec, then GO

$x^2+kx+9=2x$, where $k$ is a constant. For which value of $k$ does the equation have no real solutions?

Collect to one side, then use the discriminant.

$x^2+(k-2)x+9=0$ → $D$$=(k-2)^2-36<0$ → $-4<k<8$

A$-5$D > 0

B$7$✓

C$8$D = 0, one solution

D$10$D > 0

Move everything to one side: $x^2+(k-2)x+9=0$. No real solutions means $D<0$: $(k-2)^2-36<0$, so $-4<k<8$. Only $k=7$ fits. On Desmos you would test every k with a slider; the discriminant settles it in one step.

Or slide k on Desmos.

1

$y=x^2+(k-2)x+9$

2

$k=7$

-10

10

Move everything to one side, put k on a slider, and hunt for a value where the parabola stops crossing the x-axis. It works, but you test them one at a time. The discriminant gives $-4<k<8$ directly.

No slider will save you.

Problem · your turn: 30 sec, then GO

$-x^2+kx-16=0$, where $k$ is an integer constant. For how many values of $k$ does the equation have no real solution?

$D$$=b^2-4ac=k^2-4(-1)(-16)=k^2-64$

$k^2-64<0$ → $-8<k<8$ → $k=-7,-6,\dots,7$

A15✓

B17counts k = ±8 (D = 0)

C8forgot negative k

DInfinitely many

The discriminant is $D$$=k^2-4(-1)(-16)=k^2-64$. No real solutions means $D<0$, so $k^2<64$, i.e. $-8<k<8$. The integers $-7$ through $7$ make 15 values. A slider cannot hand you a count; the discriminant gives the whole range at once.

02

Section 2

Exponential growth.

Linear adds. Exponential multiplies.

Linear

$y=$$m$$x+$$b$

m = how much you add each step

b = the starting value

A gym: $b to join, then $m every month.

Exponential

$y=$$a$$\cdot$$b$$^{\,x}$

a = the starting value

b = the multiplier (×b each step)

$a in an account that grows ×b each year.

Pick the model.

Problem · your turn: 30 sec, then GO

A $1,000 laptop loses 15% of its value each year. Which models its value after t years?

Losing 15% keeps 85%, so multiply by $0.85$ each year.

$V=1000\cdot(0.85)^{\,t}$

A$1000(0.85)^t$✓

B$1000(1.15)^t$that is 15% growth

C$1000-150t$that is linear

D$1000(0.15)^t$keeps only 15%

Decay keeps $100\%-15\%=85\%$ each year, so the factor is $0.85$ and the model is $1000(0.85)^t$. Subtracting 150 each year (C) would be linear.

Double every day.

Problem · your turn: 30 sec, then GO

A pond has 300 algae and the amount doubles every day. How many after 3 days?

$300\cdot 2^{\,3}=300\cdot 8=$$2400$

A$1{,}800$300 × (2×3)

B$2{,}400$✓

C$900$300 × 3, added not doubled

D$600$

Doubling means $\times 2$ each day, so after 3 days $300\cdot 2^3=300\cdot 8=$$2400$. The exponent is the number of steps.

The exponent rules.

Multiply → add

$x^a \cdot x^b = x^{a+b}$

$2^2 \cdot 2^3 = 2^5$

Divide → subtract

$\dfrac{x^a}{x^b}=x^{a-b}$

$\dfrac{2^5}{2^2}=2^3$

Power of a power → multiply

$(x^a)^b=x^{ab}$

$(2^2)^3=2^6$

Zero power

$x^0=1$

$9^0=1$

Negative → flip

$x^{-a}=\dfrac{1}{x^a}$

$2^{-3}=\dfrac{1}{8}$

Fraction → root

$x^{1/n}=\sqrt[n]{x}$

$9^{1/2}=3$

Rules, or just plug in.

Problem · your turn: 30 sec, then GO

Which of the following is equivalent to $\dfrac{(x^3)^2 \cdot x^{-4}}{x}$?

Use the rules, top to bottom:

$(x^3)^2=x^6$ → $x^6 \cdot x^{-4}=x^2$ → $\dfrac{x^2}{x}=$$x$

A$x$✓

B$\dfrac{1}{x}$sign slip

C$x^3$

D$x^9$

Not sure? Plug in a number. Let $x=2$: $\dfrac{(2^3)^2 \cdot 2^{-4}}{2}=\dfrac{64 \cdot \frac{1}{16}}{2}=\dfrac{4}{2}=2$, which matches $x=2$. Both routes give $x$.

03

Section 3

Functions & graphs.

f(x) is just a machine.

f(x) is a machine. $f(3)$ means put 3 in.

$3$

→

RULE

$f(x)=x^2+1$

→

$f(3)=10$

$f(x)=0$ = the x-intercepts. $f(x)=g(x)$ = where graphs meet.

Put the number in.

Problem · your turn: 30 sec, then GO

If $f(x)=2x^2-3$, what is $f(4)$?

$f(4)=2(4)^2-3=2(16)-3=32-3=$$29$

A29✓

B13forgot to square first

C61squared 2×4

D5did 2×4−3

Put 4 in for x: $2(4)^2-3$. Square first: $4^2=16$, then $2\cdot 16-3=$$29$. Order of operations: exponent before the multiply.

Where is f(x) = 0?

Problem · your turn: 30 sec, then GO

The graph of $y=f(x)$ is shown. What are the solutions to $f(x)=0$?

A$x=-1$ and $x=3$✓

B$x=0$that is the y-value

C$x=-3$ and $x=1$sign flip

D$y=-3$the lowest point

$f(x)=0$ asks where the graph crosses the x-axis. It crosses at $x=-1$ and $x=3$, so those are the solutions.

A shifted function.

Problem · your turn: 30 sec, then GO

The table gives values of the linear function $f$. If $g(x)=f(x-2)+3$, for what value of $x$ is $g(x)=5$?

$x$	$f(x)$
1	2
2	3
3	4

$g(x)=5$ → $f(x-2)+3=5$ → $f(x-2)=2$

Table: $f=2$ when the input is $1$, so $x-2=1$ $\Rightarrow x=3$

A$x=3$✓

B$x=1$forgot the shift

C$x=-1$shifted the wrong way

D$x=5$

$g(x)=5$ means $f(x-2)+3=5$, so $f(x-2)=2$. From the table $f=2$ when the input is $1$, so $x-2=1$, giving $x=3$. The input to f is $x-2$, so solve $x-2=1$, not $x=1$.

What will you do first?

Three moves.

Before each caption appears, call out the first move in the chat.

$b^2-4ac$

Quadratics

Read the form; count with the discriminant.

$a\cdot b^{\,t}$

Exponentials

Multiply, do not add.

$f(x)$

Functions

A machine; zeros are x-intercepts.

Try three more this week.

3

Three questions, one from each theme, with worked answers.

Try → check → explain one mistake out loud.

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Math: Geometry & Trigonometry

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\(x\)	\(f(x)\)
1	2
2	3
3	4

Quadratics& Functions.

Quadratics.

A quadratic has three outfits.

The a controls the shape.

Which point is an x-intercept?

Or just let Desmos show you.

No numbers? Fundamentals win.

The Discriminant.

How many solutions?

Or graph it on Desmos.

Which k gives no solutions?

Or slide k on Desmos.

No slider will save you.

Exponential growth.

Linear adds. Exponential multiplies.

Pick the model.

Double every day.

The exponent rules.

Rules, or just plug in.

Functions & graphs.

f(x) is just a machine.

Put the number in.

Where is f(x) = 0?

A shifted function.

Three moves.

Try three more this week.

One questionbefore you go?

Quadratics
& Functions.

One question
before you go?