Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).
A) $a = - \frac { 4 } { 5 }$ , b = -8, c = -5
B) $a = - \frac { 4 } { 5 }$ , b = 8, c = 5
C) $a = \frac { 4 } { 5 }$ , b = 8, c = 5
D) $a = \frac { 4 } { 5 }$ , b = -8, c = 5

Question

Quizplus · Accepted Answer

For a quadratic function $f(x) = ax^2 + bx + c$ to have a relative maximum, $a$ must be negative. The point (5, 25) being a relative maximum means it satisfies the equation when $x = 5$, leading to $25 = a(5)^2 + b(5) + c$. The function crosses the y-axis at (0, 5), meaning $f(0) = c = 5$. To find $a$ and $b$, we use the fact that the vertex form of a parabola is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex, so here $h = 5$ and $k = 25$. The derivative $f'(x) = 2ax + b$ equals 0 at $x = 5$ for a maximum, leading to $10a + b = 0$. Solving these equations with the given points, we find $a = -\frac{4}{5}$, $b = 8$, and $c = 5$, which matches choice B.

Find Constants A, B, and C So That the Graph f(x)=ax2+bx+cf ( x ) = a x ^ { 2 } + b x + cf(x)=ax2+bx+c

Find Constants A, B, and C So That the Graph $f ( x ) = a x ^ { 2 } + b x + c$