Answer: $-2(x + 2)(x - 6)$

Build from the requirements: Roots at $x = -2$ and $x = 6$ mean the factored form includes $(x + 2)(x - 6)$. Opens downward means $a < 0$. The general form is $a(x + 2)(x - 6)$. Find the value of a using the y-intercept: Substitute $x = 0$: $a(0 + 2)(0 - 6) = a(2)(-6) = -12a$. Set $-12a = 24$, so $a = -2$. Verify all conditions: $-2(x + 2)(x - 6)$: roots at $x = -2, 6$ ✓; $a = -2 < 0$ (opens downward) ✓; y-intercept: $-2(2)(-6) = 24$ ✓. Why distractors fail: Option B has $a = 2 > 0$ (opens upward) and y-intercept $-24$. Option C has roots at $x = 2$ and $x = -6$. Option D has $a = -1$, giving y-intercept $-1(2)(-6) = 12$, not $24$.

Answer: Student A's form directly shows the y-intercept is $6$, while Student B's form directly shows the roots are $x = -1$ and $x = -3$

Verify equivalence: Expanding $2(x + 1)(x + 3) = 2(x^2 + 4x + 3) = 2x^2 + 8x + 6$. The two forms are equivalent. Identify the features each form reveals: Standard form $2x^2 + 8x + 6$ directly shows the y-intercept $c = 6$. Factored form $2(x + 1)(x + 3)$ directly shows roots at $x = -1$ and $x = -3$ (since $x + 1 = 0$ gives $x = -1$ and $x + 3 = 0$ gives $x = -3$). Why distractors fail: Option A is wrong because the y-intercept can be found from any form by substituting $x = 0$. Option B is wrong because equivalent expressions can look different. Option D is wrong because factored form reveals the roots, not the vertex.

Answer: No — the parabola opens downward, but since $k = 9 > 0$, the vertex is above the x-axis, so the parabola crosses the x-axis at two points

Analyze the student's claim: The student incorrectly links the sign of $a$ alone to the existence of x-intercepts. The sign of $a$ determines direction of opening, not by itself whether x-intercepts exist. Use vertex form to determine x-intercepts: With $a = -1 < 0$, the parabola opens downward. The vertex is at $(5, 9)$, which is above the x-axis ($k > 0$). A downward-opening parabola with its vertex above the x-axis must cross the x-axis twice. Why distractors fail: Option A is wrong because downward-opening parabolas can cross the x-axis if the vertex is above it. Option C is wrong because the sign of $a$ does affect the direction of opening, which combined with $k$ determines x-intercepts. Option D is wrong because vertex form provides enough information to determine x-intercept existence.

Answer: Standard form: $ax^2 + bx + c$

Recall what the y-intercept represents: The y-intercept is the value of the expression when $x = 0$. Why standard form is correct: In $ax^2 + bx + c$, substituting $x = 0$ immediately gives the constant $c$. No computation is needed — the y-intercept is read directly from the expression. Why distractors fail: Option A (factored form) reveals the x-intercepts (roots), not the y-intercept directly. Option C (vertex form) reveals the vertex $(h, k)$, not the y-intercept. Option D is wrong because the y-intercept requires expansion or substitution in both factored and vertex forms.

Answer: $3(x - 1)^2 - 4$

Set up the vertex form: With vertex $(1, -4)$, the expression is $a(x - 1)^2 - 4$. Use the point to find a: Substitute $(3, 8)$: $a(3 - 1)^2 - 4 = 8 \Rightarrow 4a - 4 = 8 \Rightarrow 4a = 12 \Rightarrow a = 3$. Write the final expression: The expression is $3(x - 1)^2 - 4$. Why distractors fail: Option A uses $a = 2$: $2(3 - 1)^2 - 4 = 8 - 4 = 4 \neq 8$. Option C uses $(x + 1)$ instead of $(x - 1)$, giving the wrong vertex $(-1, -4)$. Option D makes both errors.

Answer: In standard form, factored form, and vertex form

Identify the role of the leading coefficient: In all three forms — $ax^2 + bx + c$, $a(x - r_1)(x - r_2)$, and $a(x - h)^2 + k$ — the coefficient $a$ appears explicitly as the leading multiplier. Why the correct answer works: Since $a$ is visible in all three forms, you can determine the direction of opening from any representation. Why distractors fail: Options A, B, and D each restrict identification of $a$ to only one form, but all three forms include $a$ as a visible coefficient.

Answer: $(-2, 12)$

Match to the vertex form template: Vertex form is $a(x - h)^2 + k$. Rewrite the given expression: $-3(x - (-2))^2 + 12$. Thus $h = -2$ and $k = 12$. Why the correct answer works: The vertex is $(h, k) = (-2, 12)$. Why distractors fail: Option A $(2, 12)$ results from using $h = 2$ instead of $h = -2$ (sign error with the subtraction in $(x - h)$). Option C $(-2, -12)$ incorrectly negates $k$. Option D $(2, -12)$ makes both sign errors.

Answer: The conversion is valid, and the minimum value $-9$ is indeed less than $-5$

Verify the conversion: Expand $4(x - 2)^2 - 9 = 4(x^2 - 4x + 4) - 9 = 4x^2 - 16x + 16 - 9 = 4x^2 - 16x + 7$. This matches the original. Determine the minimum: Since $a = 4 > 0$, the parabola opens upward and has a minimum at the vertex. The vertex is $(2, -9)$, so the minimum value is $-9$. Evaluate the claim: $-9 < -5$ is true, so the student's conclusion is correct. Why distractors fail: Option B incorrectly states $-9$ is not less than $-5$. Option C forgets to subtract $4(4) = 16$ and add back $7$, incorrectly keeping $k = 7$. Option D subtracts $16$ instead of computing $7 - 16 = -9$.

Answer: $(3, 32)$

Find the axis of symmetry from factored form: The roots are $x = -1$ and $x = 7$. The axis of symmetry is the midpoint: $x = \frac{-1 + 7}{2} = 3$. Find the y-coordinate of the vertex: Substitute $x = 3$ into the expression: $f(3) = -2(3 + 1)(3 - 7) = -2(4)(-4) = 32$. Why the correct answer works: The vertex is at $(3, 32)$. Why distractors fail: Option B $(4, 30)$ uses an incorrect midpoint. Option C $(3, -32)$ has a sign error in the evaluation. Option D $(-3, 32)$ negates the x-coordinate of the axis of symmetry.

Answer: Both students converted correctly

Verify the vertex form: Expand $2(x - 3)^2 - 8 = 2(x^2 - 6x + 9) - 8 = 2x^2 - 12x + 18 - 8 = 2x^2 - 12x + 10$. This matches the original. Verify the factored form: Expand $2(x - 1)(x - 5) = 2(x^2 - 6x + 5) = 2x^2 - 12x + 10$. This also matches the original. Why distractors fail: Options A and B are wrong because both conversions are verified as correct. Option D is wrong because both expansions yield the original expression $2x^2 - 12x + 10$.

Answer: Different forms highlight different key features of the same parabola

Understand the purpose of equivalent forms: Equivalent forms represent the same expression and produce the same parabola. They differ only in which features are immediately visible. Why the correct answer works: Standard form highlights the y-intercept, factored form highlights the roots, and vertex form highlights the vertex. Each serves a different analytical purpose. Why distractors fail: Option A is wrong because equivalent forms graph the same parabola. Option C is wrong because no single form provides all key features more directly than the others. Option D is wrong because converting between forms preserves all properties, including roots.

Answer: $3(x - 4)(x + 2)$

Expand the vertex form: $3(x - 1)^2 - 27 = 3(x^2 - 2x + 1) - 27 = 3x^2 - 6x + 3 - 27 = 3x^2 - 6x - 24$. Factor the standard form: Factor out $3$: $3(x^2 - 2x - 8) = 3(x - 4)(x + 2)$. Why distractors fail: Option B reverses the signs of the roots: $(x + 4)(x - 2)$ gives roots $x = -4$ and $x = 2$. Option C uses $(x - 4)(x - 2)$, which expands to $x^2 - 6x + 8$, not $x^2 - 2x - 8$. Option D omits the leading coefficient $3$.

Answer: The y-coordinate of the vertex

Recall vertex form structure: In $a(x - h)^2 + k$, the vertex of the parabola is the point $(h, k)$. Why the correct answer works: The value $k$ is the y-coordinate of the vertex, representing the minimum (if $a > 0$) or maximum (if $a < 0$) value of the expression. Why distractors fail: Option A is wrong because x-intercepts are found from factored form. Option B is wrong because the y-intercept is the constant $c$ in standard form. Option D is wrong because the slope at the vertex of a parabola is always zero, and $k$ does not represent slope.

Answer: Factored form $a(x - r_1)(x - r_2)$

Identify what 'expression equals zero' means: Setting the quadratic equal to zero means finding the roots or x-intercepts. Why factored form is correct: In $a(x - r_1)(x - r_2) = 0$, the zero-product property immediately gives $x = r_1$ or $x = r_2$ with no further computation. Why distractors fail: Option A (standard form) would require factoring or using the quadratic formula. Option B (vertex form) would require algebraic manipulation to isolate $x$. Option D is wrong because standard and vertex forms require additional steps.

Answer: The x-intercepts (roots) of the quadratic expression

Understand what factored form reveals: The factored form $a(x - r_1)(x - r_2)$ is designed so that when $x = r_1$ or $x = r_2$, the entire expression equals zero. Why the correct answer works: Since the expression equals zero at $x = r_1$ and $x = r_2$, these are the roots or x-intercepts of the quadratic. Why distractors fail: Option A is wrong because the vertex is revealed by vertex form, not factored form. Option B is wrong because the y-intercept is directly shown in standard form. Option D is wrong because the extreme value is the $k$-value in vertex form.

Answer: In vertex form $a(x - h)^2 + k$, the axis of symmetry $x = h$ is read directly from the expression

Recall the axis of symmetry in vertex form: In vertex form $a(x - h)^2 + k$, the parabola's vertex is at $(h, k)$, and the axis of symmetry is the vertical line $x = h$. Why the correct answer works: The value $h$ is visible immediately in vertex form, so no computation is needed to determine $x = h$. Why distractors fail: Option A is wrong because vertex form shows the vertex, not the y-intercept, and the axis of symmetry passes through the vertex, not necessarily the y-intercept. Option C is wrong because the leading coefficient can be positive or negative in any form. Option D is wrong because the axis of symmetry can also be found from standard form (using $x = -b/(2a)$) or factored form (using $x = (r_1 + r_2)/2$).

Answer: The x-intercepts

Compare the two forms: The original $x^2 - 6x + 8$ is in standard form, which directly shows $c = 8$ (y-intercept) and $a = 1$ (opens upward). The factored form $(x - 2)(x - 4)$ directly shows roots at $x = 2$ and $x = 4$. Why the correct answer works: The x-intercepts are immediately visible in the factored form but require factoring or the quadratic formula to find from standard form. Why distractors fail: Option A: The y-intercept ($c = 8$) is already directly visible in the standard form. Option B: The direction of opening ($a = 1 > 0$) is visible in both forms. Option D: The leading coefficient ($a = 1$) is visible in both forms.

Answer: $-30$

Find the y-intercept from factored form: The y-intercept occurs when $x = 0$. Substitute into the factored form: $2(0 - 3)(0 + 5) = 2(-3)(5)$. Compute the result: $2 \times (-3) \times 5 = -30$. So the y-intercept is $-30$. Why distractors fail: Option B ($30$) results from ignoring the negative sign on the factor $(0 - 3)$. Option C ($-15$) results from forgetting the leading coefficient $2$. Option D ($15$) results from both errors combined.