Algebra 1 — Semester B Practice

Question 1 of 10

Massachusetts 8A-8BMedium Calc Diagram

For a quadratic equation ax² + bx + c = 0, the discriminant is b² − 4ac. It tells us how many times the related parabola y = ax² + bx + c crosses the x-axis — and therefore how many real solutions the equation has.

The discriminant's sign matches the number of x-axis crossings.

Which statement about D = 0 is TRUE?

ANo real solutions — the parabola doesn't reach the x-axis

BThe number of solutions can't be determined from D alone

CTwo real solutions — the parabola crosses the x-axis twice

DExactly one real solution — the parabola is tangent to the x-axis

Explanation

The discriminant b² − 4ac counts how many real solutions a quadratic equation has by reflecting how many times the parabola y = ax² + bx + c crosses the x-axis.

The three cases:
• D > 0: parabola crosses the x-axis at TWO different points → 2 real solutions.
• D = 0: parabola is *tangent* to the x-axis — it touches at exactly ONE point (the vertex itself sits on the x-axis) → 1 real solution (often called a *double root*).
• D < 0: parabola sits entirely above or below the x-axis, never touching it → 0 real solutions (the roots are complex / imaginary).

Application: D = 0 is the borderline case useful for problems like "for what value of c does ax² + bx + c = 0 have exactly one solution?" — set b² − 4ac = 0 and solve.

Question 2 of 10

Massachusetts 6A-6CMedium Word Diagram

A projectile is launched and follows the path described by the quadratic function shown below. What does the maximum point of this parabola represent?

AThe starting position of the projectile

BThe time when the projectile lands

CThe highest point reached by the projectile and the time it occurs

DThe horizontal distance traveled

Explanation

The maximum point (vertex) of the parabola represents the highest height reached by the projectile and the time at which it occurs.

Question 3 of 10

Massachusetts 11A-11BEasy Calc

A biologist models a bacterial colony where each hour, the population multiplies by x. After 3 hours the multiplier is x³, and after another 4 hours it's x⁴. The combined growth factor across all 7 hours is x³ · x⁴.

Using the product of powers property, simplify x³ · x⁴ to a single power of x.

A2x⁷ — the two factors double the coefficient

Bx⁷ — add the exponents

Cx¹² — multiply the exponents

Dx¹ — subtract the exponents

Explanation

Product of powers rule: when multiplying powers with the *same base*, add the exponents → x³ · x⁴ = x^(3+4) = x⁷.

Why the distractors are wrong: (B) Multiplying exponents (3×4=12) is the power-of-a-power rule, used only for (x³)⁴ — a single power raised to another power. (C) There's only one base, x, and no separate coefficients to double. (D) Subtracting exponents is the quotient rule (x³ ÷ x⁴), used for division, not multiplication.

Question 4 of 10

Massachusetts 7A-7CEasy Calc Word

The vertex of y = (x − 3)² + 2 is:

A(2, 3)

B(−3, 2)

C(3, −2)

D(3, 2)

Explanation

📌 Vertex form: y = a(x−h)² + k. Vertex = (h, k) = (3, 2).

Question 5 of 10

Massachusetts 7A-7CEasy Calc Word Diagram

Which graph represents a quadratic function?

BBoth

CNeither

Explanation

📌 Quadratic = U-shaped parabola. Graph A shows a parabola.
Graph B is a straight line → linear, not quadratic.

Question 6 of 10

Massachusetts 12A-12EEasy Calc

For the quadratic function f(x) = x² + 1, find f(−2).

⚠️ Pay close attention to what happens when you square a negative number.

A−3, because (−2)² = −4 and −4 + 1 = −3

B5, because (−2)² = 4 and 4 + 1 = 5

C3, because −2 + 1 + 2² = 3

D−5, because the result of any function at a negative input is negative

Explanation

Substitute x = −2 into f(x) = x² + 1: f(−2) = (−2)² + 1 = 4 + 1 = 5.

The critical step: (−2)² = (−2) × (−2) = +4, NOT −4. A negative times a negative is positive.

Common mistake that gets distractor (A): writing −2² instead of (−2)². Without parentheses, −2² is read by order of operations as −(2²) = −(4) = −4. With parentheses (−2)², you square the negative number itself, giving +4. Always write parentheses around a negative input to a power.

Question 7 of 10

Massachusetts 9A-9CEasy Word

Which represents exponential growth?

Ay = 3x + 2

By = 2ˣ

Cy = 2/x

Dy = x²

Explanation

📌 Exponential growth: y = a·bˣ where b > 1. y = 2(3)ˣ has b = 3 > 1.

Question 8 of 10

Massachusetts 10A-10EEasy Word Diagram

A rectangular vegetable garden has a length of (x + 7) feet and a width of (x + 3) feet. Which polynomial represents the area of the garden in square feet?

Ax² + 21

B2x + 10

Cx² + 10x + 21

Dx² + 4x + 21

Explanation

Area = length × width = (x + 7)(x + 3). Use FOIL: x·x + x·3 + 7·x + 7·3 = x² + 3x + 7x + 21 = x² + 10x + 21. Choice A only multiplies the first and last terms (skips the middle). C is the perimeter formula 2(L + W) wrongly applied. D mixes up the middle coefficient.

Question 9 of 10

Massachusetts 12A-12EEasy Calc Word

What is the 5th term of geometric sequence: 2, 6, 18, ...?

A162

B108

C54

D486

Explanation

📌 r = 3. a₅ = 2·3⁴ = 2·81 = 162

Question 10 of 10

Massachusetts 1A-1GEasy Calc

A square garden has an area of 64 square feet. If the side length is x feet, the relationship is x² = 64.

The algebra gives two mathematical solutions, but only one makes physical sense as a side length. Which choice gives BOTH algebraic solutions AND correctly identifies the physical side length?

Ax = 32, because half of 64 is the side length

Bx = ±8 algebraically; the garden's side length is +8 ft

COnly x = −8, because squaring a negative gives a positive

DOnly x = 8 — that's the only solution to x² = 64

Explanation

Taking the square root of both sides gives x = ±√64 = ±8, because both (+8)² = 64 AND (−8)² = 64. In pure algebra you must report both roots. For the physical garden, length can't be negative, so only +8 ft applies in context — but the algebraic answer set is {−8, +8}.

Common mistakes: (A) Dropping the negative root entirely. (D) Dividing 64 by 2 instead of taking a square root.

Tip: when you see x² = N, always write x = ±√N first, then decide which roots fit the real-world context.

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