QuestionApril 16, 2026

In the zyplane, a parabola has vertex (6,-8) and intersects the z-axis at two points. If the equation of the parabola is written in the form y=ax^2+bx+c where a, b, and c are integer constants, which of the following could be the value of a+b+c 7 A 42 A B 44 B C 46 c D (D) 48

In the zyplane, a parabola has vertex (6,-8) and intersects the z-axis at two points. If the equation of the parabola is written in the form y=ax^2+bx+c where a, b, and c are integer constants, which of the following could be the value of a+b+c 7 A 42 A B 44 B C 46 c D (D) 48
In the zyplane, a parabola has vertex (6,-8) and intersects the z-axis at two points. If the equation of the parabola is written in the form y=ax^2+bx+c where a, b, and c are integer constants, which of the following could be the value of a+b+c 7 A 42 A B 44 B C 46 c D (D) 48

Solution
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Answer

42 Explanation 1. Write equation in vertex form Vertex form: y = a(x-h)^2 + k; here h=6, k=-8, so y = a(x-6)^2 - 8. 2. Condition for intersections with z-axis z-axis means x=0. For two intersections, we check: y(0) = a(0-6)^2 - 8 = 36a - 8. 3. Ensure integer constants Expand: y = a(x^2 - 12x + 36) - 8 = ax^2 - 12a\,x + (36a - 8); b=-12a, c=36a-8. 4. Compute a+b+c a+b+c = a + (-12a) + (36a - 8) = 25a - 8. 5. Match given answer choices Choices: 42, 44, 46, 48 → 25a - 8 matches integers if a integer. 25a - 8 = 42 \Rightarrow a=2 (integer), 25a - 8 = 44 \Rightarrow a=2.08 (not integer), 25a - 8 = 46 \Rightarrow a=2.16 (not integer), 25a - 8 = 48 \Rightarrow a=2.24 (not integer). Only 42 works.

Explanation

1. Write equation in vertex form <br /> Vertex form: $y = a(x-h)^2 + k$; here $h=6$, $k=-8$, so $y = a(x-6)^2 - 8$.<br /><br />2. Condition for intersections with z-axis <br /> z-axis means $x=0$. For two intersections, we check: $y(0) = a(0-6)^2 - 8 = 36a - 8$.<br /><br />3. Ensure integer constants <br /> Expand: $y = a(x^2 - 12x + 36) - 8 = ax^2 - 12a\,x + (36a - 8)$; <br />$b=-12a$, $c=36a-8$. <br /><br />4. Compute $a+b+c$ <br /> $a+b+c = a + (-12a) + (36a - 8) = 25a - 8$.<br /><br />5. Match given answer choices <br /> Choices: 42, 44, 46, 48 → $25a - 8$ matches integers if $a$ integer. <br />$25a - 8 = 42 \Rightarrow a=2$ (integer), <br />$25a - 8 = 44 \Rightarrow a=2.08$ (not integer), <br />$25a - 8 = 46 \Rightarrow a=2.16$ (not integer), <br />$25a - 8 = 48 \Rightarrow a=2.24$ (not integer). <br />Only $42$ works.
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