Algebra 2 — Semester A Practice

Question 1 of 10

TEKS 1A-1GMedium Diagram

A poster has dimensions (x + 4) by (x + 6). Which expression represents its area?

Ax² + 10x + 24

Bx² + 10x + 10

C2x + 10

Dx² + 24

Explanation

(x + 4)(x + 6) = x² + 6x + 4x + 24 = x² + 10x + 24.

Question 2 of 10

TEKS 2A-2IEasy

The graph of f(x) = x² is shifted 3 units right and 4 units down. The new equation is:

Ag(x) = (x + 3)² − 4

Bg(x) = (x + 3)² + 4

Cg(x) = (x − 3)² + 4

Dg(x) = (x − 3)² − 4

Explanation

Right shift by h means (x − h). Down shift by k means subtract k. So f(x − 3) − 4 = (x − 3)² − 4.

Question 3 of 10

TEKS 6A-6LMedium Diagram

The graph shown is y = |x − a| + b. What are a and b?

Aa = 3, b = 3

Ba = −3, b = −3

Ca = −3, b = 3

Da = 3, b = −3

Explanation

Vertex of |x − a| + b is (a, b). Vertex shown is (3, −3), so a=3 and b=−3.

Question 4 of 10

TEKS 1A-1GHard

A rectangular pen with one shared wall uses 80 m of fencing on the other three sides. The two sides perpendicular to the wall each have width w. What value of w maximizes the area?

Aw = 40 m

Bw = 80 m

Cw = 26.67 m

Dw = 20 m

Explanation

A(w) = w(80 − 2w) = −2w² + 80w. Vertex w = −80/(−4) = 20.

Question 5 of 10

TEKS 4A-4FHard

The equation x² − (m + 2)x + (2m + 5) = 0 has two non-real complex-conjugate roots, and the product of those roots is 13. What is the sum of the roots?

C11

D−6

Explanation

For ax² + bx + c = 0 the product of the roots is c/a and the sum is −b/a (Vieta's formulas). Here a = 1, b = −(m + 2), c = 2m + 5. Product: 2m + 5 = 13, so 2m = 8 and m = 4. Sum: −b/a = m + 2 = 6. Check the conjugate condition: with m = 4 the equation is x² − 6x + 13 = 0, discriminant 36 − 52 = −16 < 0, so the roots really are non-real conjugates (3 ± 2i), and 3 + 2i plus 3 − 2i is indeed 6. The most tempting wrong answer is 4: that is the parameter m itself, reported as if solving for m finished the problem — m is only the input to the second relation, not the sum. Answering 11 comes from solving 2m + 5 = 13 by adding 5 instead of subtracting it (2m = 18, m = 9, sum 11), and −6 comes from using b/a instead of −b/a for the sum.

Question 6 of 10

TEKS 6A-6LMedium

Solve the absolute value equation |2x − 5| = 7 for all values of x.

Ano solution

Bx = 6 only

Cx = 1 or x = −6

Dx = 6 or x = −1

Explanation

|2x − 5| = 7 splits into 2x − 5 = 7 (so x = 6) and 2x − 5 = −7 (so x = −1).

Question 7 of 10

TEKS 2A-2IMedium Diagram

Which equation matches this graph?

Ay = x² − 4

By = x² + 4

Cy = (x − 4)²

Dy = (x + 4)²

Explanation

Parabola opens up with vertex on the y-axis at (0, −4) — that's y = x² shifted down by 4.

Question 8 of 10

TEKS 1A-1GEasy

$5,000 is deposited at 4% annual compound interest. The balance after t years is given by:

AA(t) = 5000(0.04)ᵗ

BA(t) = 5000 + 0.04t

CA(t) = 5000(1.4)ᵗ

DA(t) = 5000(1.04)ᵗ

Explanation

Compound interest: A = P(1 + r)ᵗ. Here P=5000, r=0.04 → A = 5000(1.04)ᵗ.

Question 9 of 10

TEKS 2A-2IHard

Let f(x) = 2x − 5. A second function g is unknown, but the composition satisfies (f ∘ g)(x) = 2x² − 8x + 3 for every real x. What is g(−1)?

A18

D13

Explanation

By the definition of composition, (f ∘ g)(x) = f(g(x)) = 2·g(x) − 5. Setting this equal to the given composite: 2·g(x) − 5 = 2x² − 8x + 3, so 2·g(x) = 2x² − 8x + 8 and g(x) = x² − 4x + 4 = (x − 2)². Then g(−1) = (−1 − 2)² = 9. Equivalently, undo f at the single point: (f ∘ g)(−1) = 2(1) + 8 + 3 = 13, and g(−1) = (13 + 5)/2 = 9. The most tempting wrong choice is 13: that is the value of the COMPOSITE at x = −1, obtained by substituting −1 into the expression handed to you instead of first stripping off the outer function f. Choosing 18 comes from adding 5 but forgetting to divide the whole expression by 2; choosing 4 comes from subtracting 5 instead of adding it when undoing f.

Question 10 of 10

TEKS 4A-4FMedium

Solve by completing the square: x² + 6x + 5 = 0.

Ax = −1 or x = −5

Bx = −1 or x = 5

Cx = −2 or x = −4

Dx = 1 or x = 5

Explanation

x² + 6x = −5 → x² + 6x + 9 = 4 → (x + 3)² = 4 → x + 3 = ±2 → x = −1 or x = −5.

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