7 Essential SAT Quadratic & Parabola Question Types (With Bluebook-Style Examples)

DIGITAL SAT • BLUEBOOK STYLE

7 Essential SAT Quadratic & Parabola Question Types

With Bluebook-Style Examples

On the Digital SAT, quadratic and parabola questions usually fall into a small number of recurring patterns. Instead of memorizing dozens of formulas, master these 7 categories and you will be prepared for the vast majority of SAT quadratic questions.

Each section below includes Bluebook-style examples, visual explanations, and step-by-step SAT strategies.

Quick SAT Roadmap

1 Vertex / Min-Max
2 X-Intercepts / Roots
3 Discriminant / Real Solutions
4 Y-Intercept Questions
5 3 Forms of a Quadratic
6 Function Transformations
7 Word Problems
CATEGORY 1 • BLUEBOOK STYLE

Vertex: Minimum & Maximum Questions

SAT quadratic questions often ask for the minimum value or maximum value of a function. The key idea is that these questions are really about the vertex of the parabola.

SATMath800 Vertex Minimum Maximum comic

🌟 The Main Idea

A quadratic function creates a parabola. Every parabola has a special point called the vertex.

f(x) = ax2 + bx + c

The vertex tells us where the graph reaches its lowest point or highest point.

That is why vertex questions are one of the most common SAT quadratic question types.

📌 The Vertex Formula

For a quadratic function in standard form

f(x) = ax2 + bx + c

the x-coordinate of the vertex is

xv = −b 2a

After finding xv, substitute it back into the function to get the y-value of the vertex.

📈 When Is It a Minimum? When Is It a Maximum?

🙂

a > 0

The parabola opens upward.

The vertex is the lowest point.

Minimum value

🙁

a < 0

The parabola opens downward.

The vertex is the highest point.

Maximum value

Important SAT Insight:

If a < 0, the parabola opens downward forever, so it cannot have a minimum value.

🎯 The #1 SAT Warning: Read the Question Carefully

Students often find the vertex’s x-coordinate and stop too early.

The SAT may ask for:

  • the x-coordinate of the vertex, or
  • the minimum/maximum value of the function.

These are different answers.

Find The X-Coordinate

xv = −b 2a

Find The Minimum Or Maximum Value

f(xv)

Plug the vertex x-value back into the function.

⚡ Quick SAT Check

If the SAT asks… Your answer is…
x-coordinate of the vertex xv
At what value of x does the minimum occur? xv
What is the minimum value? f(xv)
What is the maximum value? f(xv)
Fast SAT Strategy:
  1. Identify a.
  2. Decide whether the parabola opens up or down.
  3. Compute xv = −b 2a .
  4. Ask yourself: Do they want x or f(x)?
  5. If they want the minimum or maximum value, compute f(xv).

Detailed Worked Example

Example — Finding A Minimum Value

Suppose

f(x) = 5x2 − 30x + 73

Find the minimum value of the function.

Step 1 — Identify a, b, and c

a = 5, b = −30, c = 73

Step 2 — Find the vertex x-coordinate

xv = −(−30) 2(5)
xv = 30 10 = 3
xv = 3

Step 3 — Compute the y-value

f(3) = 5(32) − 30(3) + 73
f(3) = 45 − 90 + 73
f(3) = 28
Minimum value = 28
Since a > 0, the parabola opens upward, so the vertex gives the minimum value.
Vertex
(3, 28)
Minimum Value
28
Common SAT Mistake:

Many students stop after finding x = 3. The question asked for the minimum value, so the correct answer is 28.

BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — Vertex Questions

These original SATMath800 questions are designed to feel like official Digital SAT quadratic problems.

Question 1

What is the minimum value of

f(x) = 2x2 − 8x + 11
Show Solution
xv = −(−8) 2(2) = 2
f(2) = 2(4) − 8(2) + 11
f(2) = 3
Minimum value: 3
Question 2

What is the maximum value of

g(x) = −3x2 + 12x − 5
Show Solution

Since a = −3 < 0, the parabola opens downward, so the vertex gives the maximum.

xv = −12 2(−3) = 2
g(2) = −3(4) + 12(2) − 5
g(2) = 7
Maximum value: 7
Question 3

A parabola has vertex (4, −7) and opens upward. What is its minimum value?

Show Solution

The vertex is (4, −7).

The minimum value is the y-coordinate of the vertex.

−7
Minimum value: −7
Question 4

What is the x-coordinate of the vertex of

h(x) = x2 + 6x + 10
Show Solution
xv = −6 2(1) = −3
X-coordinate: −3

Notice that this question asks for x, not the minimum value.

Question 5

What is the minimum value of

p(x) = 4(x − 2)2 + 9
Show Solution

This is already in vertex form.

The vertex is (2, 9).

Because a = 4 > 0, the parabola opens upward.

Minimum value: 9
Fast SAT Tip:

When a question asks for the minimum value or maximum value, the answer is a y-value.

xv = −b 2a

Then compute

f(xv)

If the question asks “At what value of x…?”, stop after finding xv.

CATEGORY 2 • BLUEBOOK STYLE

X-Intercepts / Roots Questions

SAT quadratic questions frequently ask for the x-intercepts, zeros, roots, or solutions of a quadratic function. These are all different names for the same idea.

SATMath800 X-Intercepts and Roots infographic

🌟 The Big SAT Idea

An x-intercept is a point where the graph crosses the x-axis.

At every x-intercept, the y-value is 0.

y = 0

Therefore, to find the x-intercepts of a quadratic function, we set the function equal to 0.

📚 SAT Vocabulary

SAT Wording What You Do
x-intercepts Set y = 0
roots Solve the equation
zeros Solve the equation
solutions Solve the equation
values of x where f(x)=0 Solve the equation

On the SAT, these phrases usually mean exactly the same task.

🔍 Example: Find The Roots

Suppose

f(x) = x2 − 5x + 6

To find the x-intercepts, set f(x)=0:

x2 − 5x + 6 = 0

Factor the quadratic:

(x − 2)(x − 3) = 0

Use the zero-product rule:

x − 2 = 0
x − 3 = 0
x = 2, x = 3

So the graph crosses the x-axis at:

(2,0) and (3,0)

⚡ The Fastest SAT Shortcut

If a quadratic is already written in factored form, the roots can often be found in seconds.

f(x) = (x + 4)(x − 1)

Set each factor equal to zero:

x + 4 = 0 → x = −4
x − 1 = 0 → x = 1
SAT Speed Trick:

In a factor (x − a), the root is x = a.

In a factor (x + a), the root is x = −a.

⚠️ The Most Common SAT Mistake

Students often forget to change the sign.

Factor

(x − 7)
x = 7

Factor

(x + 7)
x = −7

Think: “Set the factor equal to zero and solve.”

🎯 Roots vs. X-Intercepts

Question Asks Answer Format
What are the roots? 2, 3
What are the solutions? 2, 3
What are the x-intercepts? (2,0), (3,0)
Important:

Roots are numbers.
X-intercepts are points.

⚡ Quick SAT Check

Find the roots of

x2 + x − 12 = 0

Factor:

(x + 4)(x − 3) = 0
x = −4, x = 3

Therefore:

(−4,0) and (3,0)
🚀
30-Second SAT Strategy:
  1. See the words root / zero / x-intercept / solution.
  2. Immediately set the quadratic equal to 0.
  3. Look for easy factoring first.
  4. If it factors, use the zero-product rule.
  5. Only use the quadratic formula if factoring is not obvious.

Detailed Worked Examples

Case 1 — Two Real X-Intercepts

Find the x-intercepts of

y = x2 − 7x + 10

Step 1 — Set y equal to 0

x2 − 7x + 10 = 0

Step 2 — Factor the quadratic

(x − 5)(x − 2) = 0

Step 3 — Solve each factor

x − 5 = 0 → x = 5
x − 2 = 0 → x = 2
x = 2, x = 5
The parabola crosses the x-axis at (2,0) and (5,0).

Case 2 — A Repeated Root

Find the x-intercepts of

y = x2 − 6x + 9

Step 1 — Set y equal to 0

x2 − 6x + 9 = 0

Step 2 — Factor the quadratic

(x − 3)2 = 0

Step 3 — Solve

x − 3 = 0
x = 3
The parabola touches the x-axis at (3,0) and turns around.
BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — X-Intercept Questions

Try each question before opening the solution. These are original SATMath800 questions designed to match the style and difficulty of official Digital SAT quadratics.

Question 1

What are the roots of

x2 − 9x + 20 = 0
Show Solution
(x − 5)(x − 4) = 0
x = 4, x = 5
Roots: 4 and 5
Question 2

What are the x-intercepts of

y = x2 + x − 6
Show Solution
x2 + x − 6 = 0
(x + 3)(x − 2) = 0
x = −3, x = 2
(−3,0), (2,0)
X-intercepts: (−3,0) and (2,0)
Question 3

The function

f(x) = (x − 8)(x + 2)

has which zeros?

Show Solution
x − 8 = 0 → x = 8
x + 2 = 0 → x = −2
Zeros: −2 and 8
Question 4

Which value of x is a root of

x2 − 4x − 12 = 0
Show Solution
(x − 6)(x + 2) = 0
x = 6 or x = −2
Possible roots: 6 and −2
Question 5

A quadratic has x-intercepts at (1,0) and (7,0). Which equation could represent the quadratic?

Show Solution

Roots 1 and 7 give factors

(x − 1)(x − 7)

Expand:

x2 − 8x + 7
Equation: x² − 8x + 7 = 0
Fast SAT Tip:

If a question gives the x-intercepts and , immediately write

y = (x − a)(x − b)

This is one of the fastest ways to build a quadratic equation on the SAT.

🔍
CATEGORY 3 • BLUEBOOK STYLE

Number of Real Solutions (Discriminant)

Learn the fastest SAT method for deciding whether a quadratic equation has 2, 1, or 0 real solutions.

SATMath800 How Many Real Solutions infographic

For a quadratic equation ax² + bx + c = 0, first compute the discriminant.

Δ = b² − 4ac

The sign of Δ tells you the number of real solutions without solving the equation completely.

🔑 SAT Root Formula (Used When Needed)

After computing the discriminant, the roots of ax² + bx + c = 0 can be found using the quadratic formula.

x = (−b ± √Δ) / 2a

On the SAT, you often need only the sign of Δ to determine the number of real solutions. Use the full quadratic formula only when the question asks for the actual roots.

1

Identify the Coefficients

a
coefficient of x²
b
coefficient of x
c
constant term
2

Decide Using the Discriminant

Δ > 0

Two Real Solutions

The parabola crosses the x-axis twice.

x² − 5x + 6 = 0
⇒ 2 real solutions
x = (5 ± √1) / 2
x = 2, 3

The parabola crosses the x-axis at two different points, so it has two distinct x-intercepts.

Δ = 0

One Real Solution

The parabola touches the x-axis once.

x² − 6x + 9 = 0
⇒ 1 real solution
x = 6 / 2 = 3

The parabola touches the x-axis at exactly one point. This happens when the vertex lies on the x-axis, so the parabola is tangent to the axis and the root is repeated.

Δ < 0

No Real Solutions

The parabola does not cross the x-axis.

x² + 2x + 5 = 0
⇒ 0 real solutions

Because the discriminant is negative, the square root in the quadratic formula would involve a negative number. The parabola stays above or below the x-axis and has no x-intercepts.

Detailed Worked Examples

🟢

Case 1 — Two Real Solutions

Determine the number of real solutions of x² − 5x + 6 = 0.

Step 1 — Identify a, b, and c
a = 1, b = −5, c = 6
Step 2 — Compute the discriminant
Δ = (−5)² − 4(1)(6)
Δ = 25 − 24
Δ = 1
Since Δ > 0, the equation has 2 real solutions.
Step 3 — Use the quadratic formula
x = (−(−5) ± √1) / 2(1)
x = (5 ± 1) / 2
x = 2, 3

Two real solutions mean the parabola crosses the x-axis at two distinct intercepts: (2,0) and (3,0).

🟡

Case 2 — One Real Solution

Determine the number of real solutions of x² − 6x + 9 = 0.

Step 1 — Identify a, b, and c
a = 1, b = −6, c = 9
Step 2 — Compute the discriminant
Δ = (−6)² − 4(1)(9)
Δ = 36 − 36
Δ = 0
Since Δ = 0, the equation has 1 real solution.
Step 3 — Find the repeated root
x = 6 / 2 = 3
x = 3

One real solution means the vertex is exactly on the x-axis. The parabola touches the axis at (3,0) and turns around, so the graph is tangent to the x-axis.

🔴

Case 3 — No Real Solutions

Determine the number of real solutions of x² + 2x + 5 = 0.

Step 1 — Identify a, b, and c
a = 1, b = 2, c = 5
Step 2 — Compute the discriminant
Δ = (2)² − 4(1)(5)
Δ = 4 − 20
Δ = −16
Since Δ < 0, the equation has 0 real solutions.

No real solutions mean the parabola never reaches the x-axis. The graph has no x-intercepts, so there is no real value of x that makes the expression equal to zero.

4

Quick SAT Memory Card

Discriminant Number of Real Solutions
Δ > 0 2
Δ = 0 1
Δ < 0 0
BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — SAT-Style Questions

Try each question before opening the solution. These are original SATMath800 questions designed to match the style and difficulty of official Digital SAT quadratics.

Question 1

How many real solutions does x² − 7x + 10 = 0 have?

Show Solution
Δ = (−7)² − 4(1)(10)
Δ = 49 − 40
Δ = 9
2 real solutions
x = (7 ± √9) / 2
x = 2, 5

The parabola crosses the x-axis at (2,0) and (5,0), so it has two distinct x-intercepts.

Question 2

How many real solutions does x² − 12x + 36 = 0 have?

Show Solution
Δ = (−12)² − 4(1)(36)
Δ = 144 − 144
Δ = 0
1 real solution
x = 12 / 2 = 6
x = 6

The vertex lies on the x-axis at (6,0), so the parabola touches the axis and turns around.

Question 3

How many real solutions does 2x² + 5x + 9 = 0 have?

Show Solution
Δ = 5² − 4(2)(9)
Δ = 25 − 72
Δ = −47
0 real solutions

Because the discriminant is negative, the quadratic formula would involve √(−47). The parabola has no x-intercepts.

Question 4

For what value of k does x² + 8x + k = 0 have exactly one real solution?

Show Solution

Exactly one real solution means Δ = 0.

8² − 4(1)(k) = 0
64 − 4k = 0
4k = 64
k = 16
k = 16

When k = 16, the equation becomes x² + 8x + 16 = (x + 4)², so the parabola touches the x-axis at x = −4.

Question 5

Which equation has no real solutions?

A. x² − 4x + 3 = 0
B. x² − 10x + 25 = 0
C. x² + 6x + 13 = 0
D. x² − 1 = 0
Show Solution

Check the discriminant of each choice.

A: 16 − 12 = 4
B: 100 − 100 = 0
C: 36 − 52 = −16
D: 0 − (−4) = 4
Correct answer: C

Choice C has a negative discriminant, so its parabola never crosses the x-axis and therefore has no real solutions.

Fast SAT Tip: If a question asks only for the number of real solutions, do not solve the equation completely. Compute the discriminant, check whether it is positive, zero, or negative, and choose the answer immediately.
CATEGORY 4 • BLUEBOOK STYLE

Y-Intercept Questions

SAT quadratic questions often ask for the y-intercept, f(0), or the point where the graph crosses the y-axis. This is one of the fastest SAT quadratic skills to master.

SATMath800 Y-Intercept Questions infographic

🌟 The 5-Second SAT Rule

For any quadratic in standard form

f(x) = ax2 + bx + c

the y-intercept occurs when

x = 0

Substitute x=0:

f(0) = a(0)2 + b(0) + c
f(0) = c
Memorize this:

In standard form, the constant term c is immediately the y-value of the y-intercept.

🔍 Quick Recognition Examples

Example A

y = 3x2 − 8x + 11

Here c = 11.

(0, 11)

Example B

f(x) = −2x2 + 5x − 9

Here c = −9.

(0, −9)

⚠️ The Most Common SAT Trap

Students often confuse f(0) with the y-intercept point.

Question

What is f(0)?

12

Question

What is the y-intercept?

(0, 12)

f(0) is a number.
The y-intercept is a point.

📚 Different SAT Wordings

SAT Wording Answer Format
What is the y-intercept? (0,c)
What is f(0)? c
Where does the graph cross the y-axis? (0,c)
What is the value of the function when x=0? c

These questions are usually testing the same underlying idea.

🚨 Important: Vertex Form Does NOT Show The Y-Intercept Directly

Suppose

g(x) = 2(x − 3)2 − 5

Many students incorrectly think the y-intercept is (0, −5).

You must substitute x=0:

g(0) = 2(0 − 3)2 − 5
g(0) = 2(9) − 5
g(0) = 13
(0, 13)
Key SAT Insight:

In vertex form, a quadratic is written as

f(x) = a(x − h)2 + k

The value k is the y-coordinate of the vertex.

Vertex = (h, k)

Therefore, k tells you the vertex y-value, not the y-intercept.

⚡ Why Standard Form Is Useful

Standard Form

4x2 − x + 7

The y-intercept is visible immediately.

(0, 7)

Vertex Form

4(x − 2)2 + 7

You must substitute x=0.

23

This is why SAT questions about y-intercepts often use standard form.

⚡ Quick SAT Check

Find the y-intercept of

y = −5x2 + 4x + 9

Because c=9,

f(0) = 9
(0, 9)
🚀
30-Second SAT Strategy:
  1. See y-intercept or f(0).
  2. Check whether the quadratic is in standard form.
  3. If yes, the answer is immediately the constant term c.
  4. If not, substitute x=0.
  5. Give a point for y-intercept questions and a number for f(0) questions.

Detailed Worked Examples

Case 1 — Read Directly From Standard Form

Find the y-intercept of

f(x) = 4x2 − 7x + 12

Step 1 — Identify the constant term

c = 12

Step 2 — Compute f(0)

f(0) = 12
f(0) = 12

Step 3 — Write the y-intercept as a point

(0, 12)
In standard form, the y-intercept is immediately (0,c).

Case 2 — Vertex Form Trap

Find the y-intercept of

g(x) = 2(x − 3)2 − 5

This function is in vertex form, so we must substitute x=0.

Step 1 — Substitute x = 0

g(0) = 2(0 − 3)2 − 5

Step 2 — Simplify

g(0) = 2(9) − 5
g(0) = 18 − 5
g(0) = 13

Step 3 — Write the point

(0, 13)
Important SAT Warning:

The number −5 is the vertex y-value, not the y-intercept.

BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — Y-Intercept Questions

Try each question before opening the solution. These are original SATMath800 questions designed to match the style and difficulty of official Digital SAT quadratics.

Question 1

What is the y-intercept of

y = 5x2 + 2x − 8
Show Solution

The constant term is −8.

(0, −8)
Y-intercept: (0, −8)
Question 2

Find f(0) if

f(x) = −x2 + 6x + 4
Show Solution
f(0) = 4
f(0) = 4
Question 3

What is the y-intercept of

g(x) = 3(x + 1)2 − 7
Show Solution
g(0) = 3(1)2 − 7
g(0) = 3 − 7
g(0) = −4
Y-intercept: (0, −4)
Question 4

Which quadratic has y-intercept (0, −3)?

A. x2 − 3x + 2
B. x2 + 5x − 3
C. x2 − 3x − 5
D. x2 + 3x + 3
Show Solution

The y-intercept is the constant term c.

  • A → 2
  • B → −3
  • C → −5
  • D → 3
Correct answer: B
Question 5

A parabola passes through (0, 15). What is the value of c in

y = ax2 + bx + c
Show Solution

The point (0,15) is the y-intercept.

c = 15
c = 15
Fast SAT Tip:

If a quadratic is in standard form, you can usually answer y-intercept questions without any calculation.

ax2 + bx + c

The y-intercept is simply

(0, c)

This shortcut saves valuable time on the Digital SAT.

CATEGORY 5 • BLUEBOOK STYLE

Three Forms of a Quadratic

The SAT often shows the same parabola in different forms. Strong students recognize the structure instantly instead of converting every equation into standard form.

SATMath800 Three Forms of a Quadratic infographic

🌟 The Big SAT Idea

A quadratic function can be written in three important forms. Each form reveals a different feature of the parabola most efficiently.

Form Best For
Standard: ax2 + bx + c y-intercept, discriminant
Factored: a(x − r1)(x − r2) x-intercepts / roots
Vertex: a(x − h)2 + k vertex, minimum, maximum

💡 No Need To Memorize Everything

You only need to understand three ideas:

  • Vertex: the highest or lowest point
  • X-intercepts: where y = 0
  • Y-intercept: where x = 0

The three quadratic forms are simply different windows into the same parabola. Recognizing them saves valuable time on the Digital SAT.

1️⃣ Standard Form

y = ax2 + bx + c

This is the form that appears most often on the SAT.

What It Reveals Instantly

y = 2x2 − 5x + 7

The constant term is c = 7.

(0, 7)

Best Use

  • Find f(0)
  • Find the y-intercept
  • Compute the discriminant
  • Identify a, b, and c

2️⃣ Factored Form

y = a(x − r1)(x − r2)

This form is designed for finding x-intercepts.

y = (x − 2)(x − 5)

Set each factor equal to zero:

x − 2 = 0 → x = 2
x − 5 = 0 → x = 5
(2,0) and (5,0)
Fast SAT Recognition:

Two parentheses multiplied together usually means "Think roots immediately."

3️⃣ Vertex Form

y = a(x − h)2 + k

This form reveals the vertex instantly.

y = (x − 3)2 − 4
Vertex = (3, −4)

h

x-coordinate of the vertex

k

y-coordinate of the vertex

Because the coefficient of the squared term is positive, the parabola opens upward, so −4 is the minimum value.

🔄 One Parabola, Three Different Forms

Consider the quadratic function

y = x2 − 6x + 8

Standard Form

y = x2 − 6x + 8

Y-intercept: (0, 8)

Factored Form

y = (x − 2)(x − 4)

Roots: 2 and 4

Vertex Form

y = (x − 3)2 − 1

Vertex: (3, −1)

Same graph. Different information becomes easy to see.

⚡ The 1-Second SAT Recognition Trick

See + c at the end?

Think: Y-intercept

See two parentheses multiplied?

Think: Roots / X-intercepts

See a squared parenthesis + or − a constant?

Think: Vertex / Min / Max

🎯 Quick SAT Challenge

Which form is the fastest for each question?

Question Fastest Form
Find the y-intercept Standard
Find the x-intercepts Factored
Find the minimum value Vertex
Find the vertex Vertex
Find the discriminant Standard
🚀
Real SAT Strategy:

Don't ask "How do I convert this?"

Ask "What is the question asking for?"

  • Vertex? → Vertex form
  • Roots? → Factored form
  • Y-intercept or discriminant? → Standard form

This habit saves more time than memorizing extra formulas.

Detailed Worked Examples

Case 1 — Standard → Factored Form

Rewrite the quadratic in factored form:

x2 − 5x + 6

Step 1 — Find two numbers that multiply to 6 and add to −5

(−2)(−3) = 6
−2 + (−3) = −5

Step 2 — Write the factors

x2 − 5x + 6 = (x − 2)(x − 3)

Step 3 — Read the roots

x − 2 = 0 → x = 2
x − 3 = 0 → x = 3
x = 2, x = 3
Factored form makes the x-intercepts visible immediately.

Case 2 — Standard → Vertex Form

Rewrite the quadratic in vertex form:

x2 − 6x + 5

Step 1 — Group the x terms

x2 − 6x + 5

Step 2 — Complete the square

Half of −6 is −3, and (−3)2 = 9.

x2 − 6x + 9 − 9 + 5

Step 3 — Factor the perfect square

(x − 3)2 − 4
y = (x − 3)2 − 4

Step 4 — Read the vertex

Vertex = (3, −4)

Since the coefficient of the squared term is positive, −4 is the minimum value.

Vertex form is the fastest way to find a minimum or maximum value.

Case 3 — Which Form Is Fastest?

Suppose you are asked:

What is the minimum value of y = (x − 4)2 + 1?

Recognize the form

This is already in vertex form:

y = (x − h)2 + k
Vertex = (4, 1)

Answer the actual SAT question

The question asks for the minimum value, which is the y-coordinate of the vertex.

Minimum value = 1
SAT Warning:

Many students answer 4, which is the x-coordinate. Always check whether the question asks for x or f(x).

BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — Three Forms Of A Quadratic

Try each question before opening the solution. These are original SATMath800 questions designed to match the style and difficulty of official Digital SAT quadratics.

Question 1

Which form is most useful for finding the y-intercept of

y = 2x2 − 7x + 5
Show Solution

The equation is already in standard form, and the constant term 5 gives the y-intercept immediately.

Answer: Standard Form
Question 2

Find the x-intercepts of

y = (x + 1)(x − 6)
Show Solution
x + 1 = 0 → x = −1
x − 6 = 0 → x = 6
(−1,0) and (6,0)
X-intercepts: (−1,0) and (6,0)
Question 3

What is the vertex of

y = (x + 2)2 − 9
Show Solution

Compare with

y = (x − h)2 + k

Here h = −2 and k = −9.

Vertex = (−2, −9)
Vertex: (−2, −9)
Question 4

Which equation is the factored form of

x2 − 8x + 15
Show Solution

We need two numbers that multiply to 15 and add to −8.

(−3)(−5) = 15
(x − 3)(x − 5)
Answer: (x − 3)(x − 5)
Question 5

Which equation represents the same parabola as

y = (x − 1)(x − 5)
Show Solution

Expand the factors:

x2 − 5x − x + 5
x2 − 6x + 5
Equivalent equation: x² − 6x + 5
🚀
The Real SAT Superpower:

When you see a quadratic, ask:

  • Need the vertex? → Use vertex form.
  • Need the roots? → Use factored form.
  • Need the y-intercept or discriminant? → Use standard form.

Expert SAT students choose the form that reveals the answer with the fewest steps.

CATEGORY 6 • BLUEBOOK STYLE

Function Transformations

Learn how the SAT moves a parabola left, right, up, and down without changing its basic shape. This is one of the fastest ways to answer hard Digital SAT quadratic questions.

SATMath800 Function Transformations infographic

🌟 Start With The Parent Function

Every transformation begins with the basic parabola

y = x2

Vertex

(0, 0)

Shape

Opens upward

Axis of Symmetry

x = 0

Think of this graph as the original template. Transformations simply move or stretch this template.

🧠 The Master SAT Transformation Formula

y = a(x − h)2 + k

This single formula controls almost every SAT transformation question.

Part Effect
(x − h) Shift right h units
(x + h) Shift left h units
+ k Shift up k units
− k Shift down k units
a > 1 Vertical stretch
0 < a < 1 Vertical compression
a < 0 Reflection across the x-axis

⚠️ The SAT Sign Trick

Inside the parentheses, the direction is reversed.

Right 5

(x − 5)2

Left 5

(x + 5)2
Memorize this sentence:

Minus means right. Plus means left.

📈 What Does “Shifting a Parabola” Mean?

Start with the parent function y=x². The transformed graph keeps the same basic shape, but its vertex moves to a new location.

Right 3 Units

y=(x−3)²

The vertex moves from (0,0) to (3,0).

Left 2 Units

y=(x+2)²

The vertex moves to (−2,0).

Up 4 Units

y=x²+4

The vertex moves to (0,4).

Down 3 Units

y=x²−3

The vertex moves to (0,−3).

What You Should Notice
  • Inside the parentheses changes the x-coordinate of the vertex.
  • Outside the parentheses changes the y-coordinate.
  • The basic "U" shape stays the same unless the coefficient a changes.

🔍 Quick Visual Examples

Example A

y = (x − 3)2
  • Right 3
  • No vertical shift
Vertex = (3, 0)

Example B

y = (x + 2)2 − 4
  • Left 2
  • Down 4
Vertex = (−2, −4)

Example C — Everything Together

y = −2(x − 1)2 + 5
Feature What It Means
(x − 1) Right 1
+5 Up 5
−2 Reflect downward and stretch by factor 2
Vertex = (1, 5)

⚡ Read It Without Expanding

Suppose the SAT gives you

y = −3(x + 4)2 + 2

Instead of expanding, read the information directly from the form.

Question Answer
Vertex (−4, 2)
Horizontal shift Left 4
Vertical shift Up 2
Opening direction Downward
Stretch factor 3
Key SAT Insight:

No expansion was needed. Hard-module SAT questions are often testing whether you can read the transformation directly.

🎯 Mini SAT Challenge

For each function, identify the transformation from y=x2.

Function Transformation
(x − 7)2 Right 7
(x + 3)2 Left 3
x2 + 6 Up 6
−x2 Reflect across the x-axis

🔗 Why This Connects To Vertex Questions

Transformation questions are really vertex questions in disguise.

Once you can read a(x−h)2+k quickly, you can often answer SAT questions about:

  • the vertex,
  • the axis of symmetry,
  • the minimum or maximum value,
  • the opening direction,
  • and the graph shifts

in under 10 seconds.

🚀
10-Second SAT Vertex Trick:
  1. Look for (x−h)2.
  2. Reverse the sign inside the parentheses.
  3. Keep the sign outside unchanged.
  4. Read the vertex as (h,k).
  5. Check whether a is positive or negative to decide minimum vs. maximum.

Detailed Worked Examples

Case 1 — Horizontal Shift Only

Find the vertex of

y = (x − 6)2

Step 1 — Compare with vertex form

y = (x − h)2 + k
h = 6, k = 0

Step 2 — Read the transformation

The graph of y=x2 is shifted right 6 units.

Vertex = (6, 0)
When there is no number outside the square, the vertex stays on the x-axis.

Case 2 — Horizontal + Vertical Shift

Analyze the function

y = (x + 3)2 + 8

Step 1 — Identify h and k

h = −3, k = 8

Step 2 — Determine the shifts

  • (x+3)left 3
  • up 8

Step 3 — Read the vertex

Vertex = (−3, 8)

Step 4 — Find the axis of symmetry

x = −3

Step 5 — Determine minimum or maximum

The coefficient of the squared term is positive, so the parabola opens upward.

Minimum value = 8
SAT Shortcut:

In vertex form, the axis of symmetry is always x=h.

Case 3 — Reflection + Stretch

Analyze the function

y = −2(x − 1)2 + 4

Step 1 — Read the vertex

Vertex = (1, 4)

Step 2 — Determine the opening direction

Since a = −2 < 0, the parabola is reflected across the x-axis.

Opens downward

Step 3 — Identify the stretch

The absolute value |−2|=2, so the graph is stretched vertically by a factor of 2.

Step 4 — Find the maximum value

Because the parabola opens downward, the vertex gives the maximum.

Maximum value = 4
A negative coefficient means the vertex represents a maximum, not a minimum.
BLUEBOOK-STYLE PRACTICE

🎯 Test Yourself — Function Transformations

Try each question before opening the solution. These are original SATMath800 questions designed to match the style and difficulty of official Digital SAT quadratics.

Question 1

What is the vertex of

y = (x − 4)2 + 7
Show Solution

Compare with y=(x−h)2+k.

h = 4, k = 7
Vertex = (4, 7)
Question 2

Describe the transformations from y=x2 for

y = (x + 5)2 − 2
Show Solution
  • (x+5)left 5
  • −2down 2
Left 5 units and down 2 units
Question 3

What is the maximum value of

y = −3(x − 2)2 + 1
Show Solution

The vertex is (2,1).

Since the coefficient is negative, the parabola opens downward.

Maximum value = 1
Question 4

Which equation represents a parabola shifted left 3 and up 6 from y=x2?

Show Solution

Left 3 → (x+3)2

Up 6 →

y = (x + 3)2 + 6
Answer: y = (x + 3)² + 6
Question 5

A parabola has vertex (2, −5) and opens downward. Which equation could represent it?

Show Solution

Use vertex form:

y = a(x − 2)2 − 5

To open downward, a<0.

y = −(x − 2)2 − 5
Possible equation: y = −(x − 2)² − 5
🚀
Digital SAT Transformation Shortcut:

When you see a(x−h)2+k, read the graph in this order:

  1. h → left or right shift
  2. k → up or down shift
  3. sign of a → minimum or maximum
  4. |a| → stretch or compression

This four-step process is often faster than graphing, expanding, or using a calculator.

CATEGORY 7 • BLUEBOOK STYLE

SAT Word Problems With Quadratics

Learn how the SAT uses quadratics to model projectile motion, optimization, and real-world situations. This premium lesson focuses on visual understanding first, then efficient Bluebook-style problem solving.

🚀 What Does h(t)=at²+bt+c Actually Mean?

In SAT projectile problems, the function h(t) represents the height of an object above the ground after t seconds.

h(t) = -16t2 + 80t + 6
Part Meaning
h(t) height above the ground
t time in seconds
-16t2 gravity pulls the object downward
+80t the object is launched upward
+6 the object starts 6 feet above the ground
SAT projectile problems

📈 Visualizing The Flight

🟢 Start

h(0) = 6

The ball begins 6 feet above the ground.

🟠 Highest Point

The vertex of the parabola represents the maximum height.

🔴 Landing Point

When the object reaches the ground, its height is 0.

h(t) = 0

Solving this equation gives the time when the object hits the ground.

🧩 Fully Worked SAT Projectile Example

A ball is launched from a platform. Its height is modeled by

h(t) = -16t2 + 80t + 6

Question 1: What is the maximum height?

Step 1 — Find Time Of The Vertex

t =
−b 2a
t =
−80 2(−16)
t = 2.5

Step 2 — Find The Maximum Height

h(2.5) = -16(2.5)2 + 80(2.5) + 6
h(2.5) = 106
Maximum height = 106 feet

Question 2: When does the ball hit the ground?

Step 3 — Set The Height Equal To Zero

-16t2 + 80t + 6 = 0

Solve using the quadratic formula (calculator allowed on the SAT).

t ≈ 5.1
The ball hits the ground after about 5.1 seconds.
Why Do We Ignore The Negative Root?

A negative time would mean the ball hit the ground before it was launched, which is not physically meaningful.

📐 Maximum Area Rectangle

A rectangle has perimeter 48 feet. What dimensions produce the largest possible area?

Step 1 — Write The Perimeter Equation

2x + 2y = 48
y = 24 - x

Step 2 — Write The Area Function

A(x) = x(24 - x)
A(x) = -x2 + 24x

Step 3 — Find The Vertex

x = -24 / (2(-1))
x = 12

Step 4 — Find The Other Dimension

y = 24 - 12 = 12
Dimensions = 12 ft × 12 ft
For a fixed perimeter, the rectangle with the maximum area is a square.

💰 Revenue Modeling Problem

A theater charges $20 per ticket and sells 300 tickets. For every $1 increase in the ticket price, 10 fewer tickets are sold.

Let x be the number of $1 increases.

SAT Question:

What ticket price will produce the maximum revenue, and what is that maximum revenue?

Step 1 — Revenue Model

R(x) = (20 + x)(300 - 10x)

Step 2 — Expand

R(x) = -10x2 + 100x + 6000

Step 3 — Find The Vertex

x = -100 / (2(-10))
x = 5

The revenue is maximized when x = 5, so the theater should increase the ticket price by $5.

New ticket price = $20 + $5 = $25

Step 4 — Find The Maximum Revenue

R(5) = -10(5)2 + 100(5) + 6000
R(5) = 6250
Maximum revenue = $6,250

🎯 Bluebook-Style Practice

Question 1 — Maximum Height

A projectile has height h(t) = -4t2 + 24t + 5. What is its maximum height?

A) 29
B) 36
C) 41
D) 45
Show Solution
t = -24 / (2(-4)) = 3
h(3) = -4(9) + 72 + 5 = 41
Answer: C) 41

Question 2 — Optimization

A rectangle has perimeter 60. What dimensions maximize the area?

A) 10 × 20
B) 12 × 18
C) 15 × 15
D) 5 × 25
Show Solution
Answer: C) 15 × 15

For a fixed perimeter, the maximum-area rectangle is always a square.

🏭 Production And Profit Optimization

A company manufactures custom water bottles. The profit, in dollars, from producing x bottles is modeled by

P(x) = -2x2 + 120x - 800

How many bottles should the company produce to maximize profit?

Step 1 — Recognize The Quadratic Model

The coefficient of x2 is negative, so the parabola opens downward. The maximum profit occurs at the vertex.

Step 2 — Find The Vertex

x = -120 / (2(-2))
x = 30

Step 3 — Interpret The Result

The company should produce 30 bottles.
SAT Insight:

In optimization problems, the SAT usually cares about the real-world meaning of the vertex. Here, the x-value represents the number of bottles, not the profit itself.

⚠️ Common SAT Traps

Trap 1

Finding the x-value of the vertex when the question asks for the maximum value.

Trap 2

Using a negative root for a time problem.

Trap 3

Forgetting to substitute the vertex value back into the function.

Trap 4

Confusing maximum area with maximum side length.

🏁
30-Second SAT Quadratic Strategy:
Question Asks For Use
Greatest / least value Vertex
When something becomes zero Roots
Starting amount Y-intercept
Real-world maximum / minimum Write a quadratic → Find the vertex → Interpret the answer

If you can recognize vertex, roots, y-intercept, and vertex form inside a real-world situation, you have mastered the core quadratic skills required for the Digital SAT Math section.

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