SAT Math Grade 12 — Quadratic & Exponential Functions

📐 Quadratic Functions

The most common SAT topic! Quadratic functions appear as parabolas. Know your vertex, roots, and transformations inside-out.

⚡ SUPER QUICK MEMORY POINTS

VERTEX = highest / lowest point → $(-\tfrac{b}{2a},\ f(-\tfrac{b}{2a}))$

DISCRIMINANT = $b^2-4ac$ → tells how many roots

FOIL = First Outer Inner Last → expand brackets

COMPLETE THE SQUARE → convert to vertex form

PARABOLA opens UP if $a > 0$, DOWN if $a < 0$

AXIS OF SYMMETRY = $x = -\tfrac{b}{2a}$

📌 Quadratic Formula — MUST MEMORIZE

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Standard form: $ax^2 + bx + c = 0$ | Vertex form: $y = a(x-h)^2 + k$

✏️ Worked Example — Vertex & Axis of Symmetry

Find the vertex of $f(x) = 2x^2 - 8x + 3$

Step 1: Axis of symmetry: $x = -\dfrac{b}{2a} = -\dfrac{-8}{2(2)} = \dfrac{8}{4} = 2$
Step 2: Plug back: $f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5$
✅ Vertex = $(2, -5)$

Quadratic · Vertex

What is the vertex of the parabola $f(x) = x^2 - 6x + 11$?

⚡ KEY: Use $x = -\tfrac{b}{2a}$ first, then substitute back!

💡 Explanation

$a=1,\ b=-6,\ c=11$
Axis: $x = -\dfrac{-6}{2(1)} = 3$
$f(3) = 9 - 18 + 11 = 2$
✅ Vertex = $(3, 2)$ → Answer A

Quadratic · Discriminant

How many real solutions does $3x^2 - 6x + 4 = 0$ have?

⚡ DISCRIMINANT: If $b^2 - 4ac < 0$ → NO real roots!

💡 Explanation

$\Delta = b^2 - 4ac = (-6)^2 - 4(3)(4) = 36 - 48 = -12 < 0$
Negative discriminant → NO real solutions
✅ Answer A

Quadratic · Roots / Factoring

Which values of $x$ satisfy $x^2 - 5x + 6 = 0$?

⚡ FACTORING: Find two numbers that multiply to $c$ and add to $b$

💡 Explanation

Find two numbers → multiply to 6, add to −5: that's $-2$ and $-3$
$(x-2)(x-3) = 0$
✅ $x = 2$ or $x = 3$ → Answer A

Quadratic · Vertex Form · Transformation

The function $g(x) = -(x+2)^2 + 5$ is a parabola. Which of the following is TRUE?

⚡ VERTEX FORM: $a(x-h)^2+k$ → vertex at $(h,k)$; negative $a$ = opens DOWN

💡 Explanation

$a = -1 < 0$ → opens DOWN
$h = -2,\ k = 5$ (watch sign of $h$! it's $x - (-2)$) → vertex $(-2, 5)$
✅ Answer B

Quadratic · Word Problem

A ball is thrown upward. Its height in feet after $t$ seconds is given by $h(t) = -16t^2 + 64t + 5$. What is the maximum height the ball reaches?

⚡ MAX HEIGHT = vertex's y-value → find $t = -\tfrac{b}{2a}$ first

💡 Explanation

$a=-16,\ b=64$
$t = -\dfrac{64}{2(-16)} = -\dfrac{64}{-32} = 2$ seconds
$h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69$
✅ Maximum height = 69 feet → Answer B

Quadratic · Complete the Square

Which is the vertex form of $f(x) = x^2 + 4x - 1$?

⚡ COMPLETE THE SQUARE: Add & subtract $\left(\tfrac{b}{2}\right)^2$ inside!

💡 Explanation

$x^2 + 4x - 1$
Add/subtract $\left(\tfrac{4}{2}\right)^2 = 4$:
$= x^2 + 4x + 4 - 4 - 1$
$= (x+2)^2 - 5$
✅ Answer A

📈 Exponential Functions

Exponential functions GROW or DECAY super fast. SAT loves growth/decay word problems and comparing exponential vs linear!

⚡ SUPER QUICK MEMORY POINTS

GROWTH: $f(x) = a \cdot b^x$, where $b > 1$

DECAY: $f(x) = a \cdot b^x$, where $0 < b < 1$

INITIAL VALUE = $a$ (when $x=0$)

NEVER TOUCHES x-axis → asymptote at $y=0$

DOUBLE → multiply by 2; HALF → multiply by $\tfrac{1}{2}$

LOG undoes EXPONENTIAL — inverse functions!

📌 Growth & Decay Formulas — MUST MEMORIZE

\[ A(t) = A_0 \cdot b^{t/T} \]

$A_0$ = initial amount | $b$ = growth/decay factor | $T$ = period | Percent rate: $b = 1 + r$ (growth) or $b = 1 - r$ (decay)

✏️ Worked Example — Exponential Growth

A bacteria colony starts at 500 and doubles every 3 hours. How many after 9 hours?

Model: $A(t) = 500 \cdot 2^{t/3}$
Plug in $t=9$: $A(9) = 500 \cdot 2^{9/3} = 500 \cdot 2^3 = 500 \cdot 8 = 4000$
✅ Answer: 4,000 bacteria

Exponential · Growth vs Decay

Which function represents exponential decay?

⚡ DECAY: base $b$ must be between 0 and 1 → $0 < b < 1$

💡 Explanation

A → $b = 2 > 1$: growth ✗
B → $b = 1.2 > 1$: growth ✗
C → $b = 0.7$, and $0 < 0.7 < 1$: DECAY ✅
D → linear, not exponential ✗
✅ Answer C

Exponential · Initial Value

The function $P(t) = 200 \cdot (1.05)^t$ models a population. What was the initial population when $t = 0$?

⚡ INITIAL VALUE: Plug $x = 0$ → any base to power 0 = 1!

💡 Explanation

$P(0) = 200 \cdot (1.05)^0 = 200 \cdot 1 = 200$
Remember: anything to the power 0 equals 1!
✅ Answer C

Exponential · Halving / Decay Word Problem

A radioactive substance has a half-life of 5 years. If you start with 80 grams, how much remains after 15 years?

⚡ HALF-LIFE: Every period → multiply by $\tfrac{1}{2}$. Count how many periods!

💡 Explanation

15 years ÷ 5 years per period = 3 half-life periods
$A = 80 \cdot \left(\dfrac{1}{2}\right)^3 = 80 \cdot \dfrac{1}{8} = 10$ grams
✅ Answer C

Q10

Exponential · Equation Solving

Solve for $x$: $2^{x+1} = 32$

⚡ SAME BASE TRICK: Write both sides as powers of the same base, then equate exponents!

💡 Explanation

$32 = 2^5$ → so $2^{x+1} = 2^5$
Same base → equate exponents: $x + 1 = 5$
$x = 4$
✅ Answer B

Q11

Exponential · Percent Growth Rate

A savings account grows at 6% per year. If $\$1{,}000$ is deposited, which expression gives the amount after $t$ years?

⚡ PERCENT GROWTH: Rate $r\%$ → base $= 1 + \tfrac{r}{100}$

💡 Explanation

6% growth → base $= 1 + 0.06 = 1.06$
Initial amount $= 1000$
A → base 0.06 is decay, wrong ✗
B → linear, not exponential ✗
C → $1000 \cdot (1.06)^t$ ✅
✅ Answer C

Q12

Exponential · Tricky — Negative Exponent

What is the value of $4^{-2}$?

⚡ NEGATIVE EXPONENT: $a^{-n} = \dfrac{1}{a^n}$ — flip it!

💡 Explanation

$4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}$
Negative exponent = reciprocal. Never negative value!
✅ Answer C

🎯 Mixed Challenge Problems

These combine multiple concepts — quadratic, exponential, and bonus topics. The sneaky ones that trip students up!

⚡ TRICKY TRAPS — WATCH OUT!

TRAP 1: Vertex form sign flip — $(x+2)^2$ → vertex at $x = -2$, NOT $+2$!

TRAP 2: Negative exponent ≠ negative number

TRAP 3: "Maximum" in real-life → parabola opening DOWN

TRAP 4: Linear vs exponential — constant DIFFERENCE = linear; constant RATIO = exponential

TRAP 5: $b^0 = 1$ always — any base! ($0^0$ is special, not on SAT)

Q13

Quadratic · Intersections

At how many points does $y = x^2 - 4$ intersect the x-axis?

⚡ X-INTERCEPTS: Set $y = 0$, then count solutions using discriminant!

💡 Explanation

Set $x^2 - 4 = 0$ → $x^2 = 4$ → $x = \pm 2$
Two x-intercepts: $(2,0)$ and $(-2,0)$
✅ Answer C

Q14

Exponential · Comparing Tables

A table shows: $x = 0, 1, 2, 3$ and $y = 3, 6, 12, 24$. Is this relationship linear or exponential?

⚡ CHECK RATIO: Divide each term by the previous. Constant ratio = exponential!

💡 Explanation

Differences: $6-3=3,\ 12-6=6,\ 24-12=12$ → NOT constant → not linear
Ratios: $\frac{6}{3}=2,\ \frac{12}{6}=2,\ \frac{24}{12}=2$ → constant ratio of 2 ✅
Model: $y = 3 \cdot 2^x$
✅ Answer C

Q15

Quadratic · Systems of Equations

Which value of $k$ makes $x^2 - kx + 9 = 0$ have exactly one real solution?

⚡ ONE SOLUTION: Discriminant = 0 → $b^2 - 4ac = 0$. Solve for $k$!

💡 Explanation

For one solution: $\Delta = 0$
$(-k)^2 - 4(1)(9) = 0$
$k^2 - 36 = 0 \Rightarrow k^2 = 36 \Rightarrow k = 6$ (positive value)
✅ Answer B

Q16

Exponential · Fractional Exponents

Simplify: $27^{2/3}$

⚡ FRACTIONAL EXPONENT: $a^{m/n} = \left(\sqrt[n]{a}\right)^m$ → root first, then power!

💡 Explanation

$27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$
Step 1: Cube root of 27 = 3
Step 2: Square it: $3^2 = 9$
✅ Answer C

Q17

Quadratic · Sum & Product of Roots

For $2x^2 - 10x + 8 = 0$, what is the product of the roots?

⚡ VIETA'S FORMULAS: Sum of roots = $-\tfrac{b}{a}$, Product = $\tfrac{c}{a}$. No solving needed!

💡 Explanation

Product of roots $= \dfrac{c}{a} = \dfrac{8}{2} = 4$
(Verify: roots are $x=1, x=4$, product = $1 \times 4 = 4$ ✓)
✅ Answer A

Q18

Exponential · Graph Interpretation

The graph of $f(x) = 3^x$ is shifted 2 units left and 1 unit down. What is the new equation?

⚡ SHIFTS: Left $h$ → $(x+h)$; Down $k$ → subtract $k$ outside. Inside = horizontal, outside = vertical!

💡 Explanation

Left 2: replace $x$ with $(x+2)$ → $3^{x+2}$
Down 1: subtract 1 outside → $3^{x+2} - 1$
Common trap: left shift = PLUS inside (not minus!)
✅ Answer C

Q19

Quadratic · Area / Word Problem

A rectangular garden has length $(x+5)$ and width $(x-2)$. If the area is 40 sq ft, which equation can be used to find $x$?

⚡ AREA = length × width. Set up the equation, then bring to standard form!

💡 Explanation

Area: $(x+5)(x-2) = 40$
FOIL: $x^2 - 2x + 5x - 10 = 40$
$x^2 + 3x - 10 = 40$
$x^2 + 3x - 50 = 0$
Check: $x=5$: $5^2+15-50=0$ ✅
✅ Answer A (and C — but A is the equation form; C gives extra unnecessary info so A is cleaner)

Q20

Exponential + Quadratic · Mixed Challenge ⭐

A population is modeled by $P(t) = 500 \cdot 2^t$. At the same time, a linear model predicts $L(t) = 1000t + 500$. At $t = 0$, which is larger, and by how much?

⚡ PLUG IN $t = 0$ to both functions! Which gives the bigger output?

💡 Explanation

$P(0) = 500 \cdot 2^0 = 500 \cdot 1 = 500$
$L(0) = 1000(0) + 500 = 0 + 500 = 500$
They are exactly equal at $t=0$! Both start at 500.
(The exponential grows much faster for $t > 0$)
✅ Answer C

Quadratic & ExponentialFunctions Workbook