Question

The quadratic function f(x)=ax^2+4x+c has a maximum at the point (1, 9) . What are the values of a and c?

Answer 1

Answer: a = -2  and  c = 7

Step-by-step explanation:

(this answer uses some basic calculus)

We know that $f'(x) = 2ax + 4$, and since $(1,9)$ is a maximum of $f$, then $f'(1) = 0$. That means $2a + 4 = 0$, so $\boxed{a = -2}$. Then since $f(1) = 9$, we have $-2(1)^2 + 4(1) + c = 9$, so $\boxed{c = 7}$.

Answer 2

Answer:

a = -2

c = 7

Step-by-step explanation:

Given the quadratic function, f(x) = ax² + 4x + c, for which its vertex is the maximum point occurring at (1, 9), and b = 4.

Solve for the value of a:

Since the x-coordinate of the vertex, x = 1, can be calculated using the formula, $x = \frac{-b}{2a}$:

We can substitute the value of the x-coordinate and b = 4 into the formula, and solve for the value of a:

$x = \frac{-b}{2a}$

$1 = \frac{-4}{2a}$

Multiply both sides by 2a:

(2a) 1 =  $\frac{-4}{2a} (2a)$

2a = -4

Divide both sides by 2 to solve for a:

$\frac{2a}{2} = \frac{-4}{2}$

a = -2

Therefore, the value of a = -2.

Solve for the value of c:

Next, to solve for c, substitute the coordinate values of the vertex, (1, 9) into the given quadratic function:

f(x) = ax² + 4x + c

9 = -2(1)² + 4(1) + c

9 = -2 + 4 + c

9 = 2 + c

Subtract 2 from both sides to isolate c:

9 - 2 = 2 - 2  + c

7 = c

Double-check:

In order to double-check the validity of our values for a and c, substitute a = -2, and c = 7 into the function, along with the coordinates of the vertex, (1, 9):

f(x) = -2x² + 4x + 7

9 = -2(1)² + 4(1) + 7

9 = -2 + 4 + 7

9 = 9 (True statement).

Therefore, the correct answers are: a = -2, and c = 7.